Wikiversity:Requests for Deletion

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[[Category:Wikiversity deletion]]

__TOC__

== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

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

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

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

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

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

Prefix index (max 200 items?):
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

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

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

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

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

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

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

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

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

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

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

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

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

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

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

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

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)

== [[Pragmatics/History]] ==
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

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

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

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

Archive

== [[Gravitational torsion field]] ==
{{archive top|I have gone ahead and deleted this. I don’t see much point in moving to userspace as the users currently inactive. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:30, 27 March 2026 (UTC)}}
The article [[Gravitational torsion field]] is proposed for deletion. Firstly, this article has no relation to the gravitational torsion field described in the article [[Physics/Essays/Fedosin/Gravitational torsion field]]. Secondly, the article's content is a mishmash of unrelated ideas and assumptions, many of which are not even related to gravitation.
[[User:Fedosin|Fedosin]] ([[User talk:Fedosin|обсуждение]] • [[Special:Contributions/Fedosin|вклад]]) 12:38, 9 November 2025 (UTC)
: '''Move to user space''', which is quasi-deletion. Searching the article for &quot;Gravitational torsion field&quot; finds nothing, not in the text, not in the references. The article is not labeled as original research, yet the headword &quot;Gravitational torsion field&quot; does not trace anywhere (it cannot trace anywhere from the body text since the body text does not have the headword). These are red flags. Further reading: [[W:User_talk:Swbraithwaite]], [[W:User talk:SWBPAUSEWATCH]], more red flags. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:48, 9 November 2025 (UTC)
:'''Delete'''. Low quality. Out of scope. Author no longer active on Wikiversity and has problematic WMF editing history. More detail: [https://chatgpt.com/share/6911338b-99ac-8008-833a-fb64e569a010 ChatGPT review]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:40, 10 November 2025 (UTC)
:: I think we should move to user space unless we have a specific reason to outright delete, consistent with the position taken rather passionately by Guy vandegrift and supported by some other people, including probably by Dave Braunschweig who often moved pages to user space. Moreover, whether the page is out of scope, I am not sure; we do have author-specific articles (e.g.  [[Physics/Essays/Fedosin/Gravitational torsion field]]) and if the page was solid enough, it would not be out of scope, I think. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:33, 10 November 2025 (UTC)
:::Wikiversity is not free hosting service. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:47, 11 March 2026 (UTC)
:'''Delete'''. I dont understand its conntent, but the major obstacle is how to use this conentent. It looks like the copy of Wikipedia article so I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:47, 11 March 2026 (UTC)

May be it is a simplest variant for the case.[[User:Fedosin|Fedosin]] ([[User talk:Fedosin|обсуждение]] • [[Special:Contributions/Fedosin|вклад]]) 14:10, 9 November 2025 (UTC)
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== [[IMHA Research Archives]] ==
I propose to '''move to userspace''', including the subpages. I struggle to understand how Wikiversity readers are supposed to benefit from the material here and in the subpages. In the log, there is e.g. '10 February 2019 Marshallsumter discuss contribs deleted page IMHA Research Archives (content was: &quot;{&lt;nowiki/&gt;{Delete|Author request}} Thanks! -&quot;)', so the page was deleted before, but not the subpages.

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

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

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

==[[/PubMed/]]==

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

==[[/Zotero/]]==

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

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

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

== [[Fairy Rings]] ==
{{archive top|Deleted, per consensus}}
The page and subpages do not show anything useful; this has been so since 2007, I think (maybe I do not concentrate). Author: [[User:Juandev]]. '''Move to user space''' (or delete if preferred by the author and co-authors?). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:57, 18 November 2025 (UTC)

For instance, [[Fairy Rings/Database/Lublaňská 25]] was created in 2014 by [[User:Juandev (usurped)]]; there are lat-lon coordinates and an empty section for observations.

In [[Fairy Rings/Database]], I entered auto subpage generation. It found:
* [[Fairy Rings/Database/Lublaňská 25]]
* [[Fairy Rings/Database/Test]]
* [[Fairy Rings/Database/Test 2]]
* [[Fairy Rings/Database/Test 2/May 14, 2014]]
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:02, 18 November 2025 (UTC)

The project has an introduction to the issue and clearly stated instructions. I don't see the lack of participation in the project yet as a problem. Wikiversity is not Wikipedia, we are not aiming for pages full of text here, however, if someone is bothered by it, it can be deleted. For me, it would be enough to edit and update the project a little. --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:40, 20 November 2025 (UTC)

'''Keep'''. Clear objective that is in scope. '''Delete''' the test database pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:01, 22 November 2025 (UTC)
: [[WV:Deletion]] indicates that pages for which &quot;learning outcomes are scarce&quot; (as is the case here) are to be deleted. I don't see any policy or guideline indicating that something having a clear objective that is in scope of the English Wikiversity is alone grounds for keeping, regardless of how useless or underdeveloped the page is (perhaps I was not looking carefully enough). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:29, 12 December 2025 (UTC)
:Thats a good point. I would '''delete''' test pages which I have created and I would '''keep''' the rest. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:52, 11 March 2026 (UTC)
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== [[Palliative medicine]] ==

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

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)

== [[Theory of Everything (From Scratch) Project]] ==
{{archive top|Deleted [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:19, 3 April 2026 (UTC)}}
Underdeveloped project since 2010. Original author has been inactive wiki-wide since then. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 01:45, 1 January 2026 (UTC)

:Yup, I guess we can delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:57, 16 March 2026 (UTC)
:@[[User:Atcovi|Atcovi]] @[[User:Juandev|Juandev]] Does this include, [[Theory of Everything (From Scratch) Project/The Origin]]? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 28 March 2026 (UTC)
::Yes as its low-quality, is part of the project, has not been improved on since 2010. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:43, 31 March 2026 (UTC)
::Yes, the tradition is, that it includes all subpages if it is not stated otherwise. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:41, 1 April 2026 (UTC)
:{{done}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:18, 3 April 2026 (UTC)
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== [[Seven Heavens]] ==
{{archive top|moved to userspace. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:23, 2 April 2026 (UTC)}}
Seems to be someone's personal beliefs rather than educational content that reflects Wikiversity's learning policies. It is not even labeled as such either. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:36, 19 January 2026 (UTC)

:This seems like '''speedy delete''' material to me. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&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; 22:10, 19 January 2026 (UTC)
:Agree [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:26, 19 January 2026 (UTC)
::Moved to userspace. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:23, 2 April 2026 (UTC)
{{archive bottom}}

== [[Peace studies]] ==
{{archive top|'''Deleted''' per consensus.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:23, 27 March 2026 (UTC)}}
Underdeveloped since 2006/2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:39, 21 January 2026 (UTC)
:'''Delete''' —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&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; 19:22, 21 January 2026 (UTC)
:Delete [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:15, 22 January 2026 (UTC)

{{archive bottom}}

== [[Canadian Wilderness]] ==
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

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

== [[Systemic Lupus Erythematosus]] ==
{{archive top|'''Deleted''' - AI-slop with no educational objectives}}
Clearly seems like an ai-generated article and it seems to be out of Wikiversity’s scope. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 10:08, 11 March 2026 (UTC)

:'''Delete''', copy of Wikipedia article. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:28, 27 March 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

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

== False flag &quot;authority hack&quot; user page deletion ==

'''Undeletion requested'''

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

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

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

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

Here is my answer I posted to the discussion today: 

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

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


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

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

Best regards, 

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

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

== Undeletion request ==

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

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

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)</text>
      <sha1>084v7a6ocgqc7a3pzu7hzkwz2t89k2r</sha1>
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[[Category:Wikiversity deletion]]

__TOC__

== [[Korean/Words]] ==
(I go to RfD instead of ''proposed deletion'' since many pages are affected.)

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

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

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

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

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

Prefix index (max 200 items?):
{{Small START}}
{{Special:Prefixindex/Korean/Words}}
{{Small END}}

--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:33, 24 September 2025 (UTC)

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

== [[Enhancing Web Browser Security through Cookie Encryption]] ==
To avoid further conflict with the user who entered this text into Wikiversity, I am opening a RFD request.

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

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

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

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

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

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

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

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

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

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

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

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

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

'''Keep'''. This is a legitimate student learning project that may be of use to others. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 22 November 2025 (UTC)

== [[Pragmatics/History]] ==
Another KYPark page and subpages with unclear organization scheme. Contains fairly many redlinked items. See also [[User:KYPark/Literature]], perhaps a similar concept. Unlikely to be really useful for others but KYPark. '''Move to user space'''.

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

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

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

Archive

== [[Gravitational torsion field]] ==
{{archive top|I have gone ahead and deleted this. I don’t see much point in moving to userspace as the users currently inactive. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 12:30, 27 March 2026 (UTC)}}
The article [[Gravitational torsion field]] is proposed for deletion. Firstly, this article has no relation to the gravitational torsion field described in the article [[Physics/Essays/Fedosin/Gravitational torsion field]]. Secondly, the article's content is a mishmash of unrelated ideas and assumptions, many of which are not even related to gravitation.
[[User:Fedosin|Fedosin]] ([[User talk:Fedosin|обсуждение]] • [[Special:Contributions/Fedosin|вклад]]) 12:38, 9 November 2025 (UTC)
: '''Move to user space''', which is quasi-deletion. Searching the article for &quot;Gravitational torsion field&quot; finds nothing, not in the text, not in the references. The article is not labeled as original research, yet the headword &quot;Gravitational torsion field&quot; does not trace anywhere (it cannot trace anywhere from the body text since the body text does not have the headword). These are red flags. Further reading: [[W:User_talk:Swbraithwaite]], [[W:User talk:SWBPAUSEWATCH]], more red flags. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:48, 9 November 2025 (UTC)
:'''Delete'''. Low quality. Out of scope. Author no longer active on Wikiversity and has problematic WMF editing history. More detail: [https://chatgpt.com/share/6911338b-99ac-8008-833a-fb64e569a010 ChatGPT review]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:40, 10 November 2025 (UTC)
:: I think we should move to user space unless we have a specific reason to outright delete, consistent with the position taken rather passionately by Guy vandegrift and supported by some other people, including probably by Dave Braunschweig who often moved pages to user space. Moreover, whether the page is out of scope, I am not sure; we do have author-specific articles (e.g.  [[Physics/Essays/Fedosin/Gravitational torsion field]]) and if the page was solid enough, it would not be out of scope, I think. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:33, 10 November 2025 (UTC)
:::Wikiversity is not free hosting service. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:47, 11 March 2026 (UTC)
:'''Delete'''. I dont understand its conntent, but the major obstacle is how to use this conentent. It looks like the copy of Wikipedia article so I would delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:47, 11 March 2026 (UTC)

May be it is a simplest variant for the case.[[User:Fedosin|Fedosin]] ([[User talk:Fedosin|обсуждение]] • [[Special:Contributions/Fedosin|вклад]]) 14:10, 9 November 2025 (UTC)
{{archive bottom}}

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

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

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

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

==[[/PubMed/]]==

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

==[[/Zotero/]]==

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

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

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

== [[Fairy Rings]] ==
{{archive top|Deleted, per consensus}}
The page and subpages do not show anything useful; this has been so since 2007, I think (maybe I do not concentrate). Author: [[User:Juandev]]. '''Move to user space''' (or delete if preferred by the author and co-authors?). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:57, 18 November 2025 (UTC)

For instance, [[Fairy Rings/Database/Lublaňská 25]] was created in 2014 by [[User:Juandev (usurped)]]; there are lat-lon coordinates and an empty section for observations.

In [[Fairy Rings/Database]], I entered auto subpage generation. It found:
* [[Fairy Rings/Database/Lublaňská 25]]
* [[Fairy Rings/Database/Test]]
* [[Fairy Rings/Database/Test 2]]
* [[Fairy Rings/Database/Test 2/May 14, 2014]]
--[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:02, 18 November 2025 (UTC)

The project has an introduction to the issue and clearly stated instructions. I don't see the lack of participation in the project yet as a problem. Wikiversity is not Wikipedia, we are not aiming for pages full of text here, however, if someone is bothered by it, it can be deleted. For me, it would be enough to edit and update the project a little. --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:40, 20 November 2025 (UTC)

'''Keep'''. Clear objective that is in scope. '''Delete''' the test database pages. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:01, 22 November 2025 (UTC)
: [[WV:Deletion]] indicates that pages for which &quot;learning outcomes are scarce&quot; (as is the case here) are to be deleted. I don't see any policy or guideline indicating that something having a clear objective that is in scope of the English Wikiversity is alone grounds for keeping, regardless of how useless or underdeveloped the page is (perhaps I was not looking carefully enough). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:29, 12 December 2025 (UTC)
:Thats a good point. I would '''delete''' test pages which I have created and I would '''keep''' the rest. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:52, 11 March 2026 (UTC)
{{archive bottom}}

== [[Palliative medicine]] ==

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

:Delete, per nominator [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:16, 22 January 2026 (UTC)
:Yes, I would also expect there to be more and especially that someone would write how to use it. However, it still seems to me to be a useful thing in the sense of a syllabus, so that someone who is interested in the topic knows what information to obtain in order to get a complete picture of the topic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:55, 16 March 2026 (UTC)

== [[Theory of Everything (From Scratch) Project]] ==
{{archive top|Deleted [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:19, 3 April 2026 (UTC)}}
Underdeveloped project since 2010. Original author has been inactive wiki-wide since then. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 01:45, 1 January 2026 (UTC)

:Yup, I guess we can delete it. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:57, 16 March 2026 (UTC)
:@[[User:Atcovi|Atcovi]] @[[User:Juandev|Juandev]] Does this include, [[Theory of Everything (From Scratch) Project/The Origin]]? [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 28 March 2026 (UTC)
::Yes as its low-quality, is part of the project, has not been improved on since 2010. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:43, 31 March 2026 (UTC)
::Yes, the tradition is, that it includes all subpages if it is not stated otherwise. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:41, 1 April 2026 (UTC)
:{{done}} [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:18, 3 April 2026 (UTC)
{{archive bottom}}

== [[Seven Heavens]] ==
{{archive top|moved to userspace. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:23, 2 April 2026 (UTC)}}
Seems to be someone's personal beliefs rather than educational content that reflects Wikiversity's learning policies. It is not even labeled as such either. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:36, 19 January 2026 (UTC)

:This seems like '''speedy delete''' material to me. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&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; 22:10, 19 January 2026 (UTC)
:Agree [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:26, 19 January 2026 (UTC)
::Moved to userspace. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:23, 2 April 2026 (UTC)
{{archive bottom}}

== [[Peace studies]] ==
{{archive top|'''Deleted''' per consensus.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:23, 27 March 2026 (UTC)}}
Underdeveloped since 2006/2007. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 17:39, 21 January 2026 (UTC)
:'''Delete''' —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&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; 19:22, 21 January 2026 (UTC)
:Delete [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 11:15, 22 January 2026 (UTC)

{{archive bottom}}

== [[Canadian Wilderness]] ==
This page doesn't seem to belong to wikiversity. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:55, 6 February 2026 (UTC)

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

== [[Systemic Lupus Erythematosus]] ==
{{archive top|'''Deleted''' - AI-slop with no educational objectives}}
Clearly seems like an ai-generated article and it seems to be out of Wikiversity’s scope. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 10:08, 11 March 2026 (UTC)

:'''Delete''', copy of Wikipedia article. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:28, 27 March 2026 (UTC)
{{archive bottom}}

== [[LQR Control for an Inverted Pendulum]] ==
Underdeveloped resource, has not been edited for more than a decade. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 08:03, 16 March 2026 (UTC)

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

== False flag &quot;authority hack&quot; user page deletion ==

'''Undeletion requested'''

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

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

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

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

Here is my answer I posted to the discussion today: 

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

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


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

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

Best regards, 

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

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

== Undeletion request ==

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

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

:Page: [[Is it likely that Earth has been visited by aliens millions of years ago?]]
:Ping: [[User:Atcovi|Atcovi]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:21, 11 April 2026 (UTC)
:There is no need for a discussion for straight garbage-level, pseudoscientific content.
:For '''Is it likely that Earth has been visited by aliens millions of years ago?''', the flaws for this page wouldn't even take someone more than a few minutes to assess:
:* Essentially, the &quot;pro&quot; arguments unproven claims being derived from irrelevant, established facts (basically: &quot;it is likely aliens have came because Earth has existed for so long [sources proving Earth's longevity]&quot;). These are not serious, scientifically-backed arguments - these are non sequiturs. It's as if I said Wikipedia has existed longer than my existence on Earth ([https://d1wqtxts1xzle7.cloudfront.net/74351725/eyJoIjogImZiODhmYzNkODU1N2UxMWExYzUyODJiYzgzZTRmZDM4OTBjODY5YWMzMjA3NDNmOWEyZTA0ZTU3ZGYwZjAyYTkiLCAidSI6ICJodHRwczovL3B1cmUuaHZhLm5sL3dz-libre.pdf?1636354596=&amp;response-content-disposition=inline%3B+filename%3DCritical_Point_of_View_A_Wikipedia_Reade.pdf&amp;Expires=1775872055&amp;Signature=GqbUZboYRvUYWi~aW40LT5eZSHrLuDL3o0-DxAH8vSvcJcGAuyByZWLF2oHTY6GlB72TqvZxpE-v9d4gvsA6myriYqO~QQQZgWxjT2JXjUWC-yiPcTF4l~lroJSi4dY0v9eKiBcU03l-aeUdrX8~UPfi0TfW0IhsmzH-VBR6X6FrzRpIqc6uM6n9YXfr5FRB3aCqqokU690af3n0Hguaub1Zgmh9qjYYqzBS0VOOHjKTTEQnDuadX3jl5CQeXYTaeCC3H0hMeVwHlratbrnuFEKC1aN0-5znCUoSzMEg21ECzGPTrSDM1W05dcK-u0ZTCeUGKAuC-2yRFL3sY46MIw__&amp;Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA#page=157 reputable source proving ''this'' fact]), therefore it's likely that my birth took place solely for the sake of me experiencing Wikipedia (0 backing). It makes no sense and no person with at least a high school-level of intelligence would take this seriously. 
:* What is worse is that the user is being misleading with their &quot;[the page is] sourced very well&quot; claim. The sources ''themselves'' don't even back up the claims. It's just used as proof for an established concept, where the user then uses this established concept to jump to an unsupported, laughable conclusion that is pulled out of thin air. It's utterly ridiculous to even consider such a page for mainspace since it clearly violates our [[Wikiversity:Verifiability]] policy. This is, once again, pseudoscientific content that has caused our website to reduce in quality over the last few years. 
:* Going source by source, we can see that:
:#[https://web.archive.org/web/20250918011642/https://timesofindia.indiatimes.com/blogs/thebigd/compress-earths-history-into-24-hrs-humans-came-at-1158-pm-yet-killed-70-of-wildlife/ ‘Compress Earth’s history into 24 hrs. Humans came at 11:58 pm, yet killed 70% of wildlife’] is literally just a blog post which doesn't even mention aliens or extraterrestrial life. It just talks about Earth's history in accordance with the 24-hour metric of time, and the author tries to use this article as a 'piece in the puzzle' of aliens &quot;possibly&quot; visiting Earth... which, once again, is unsupported and is not backed up anywhere in the article.
:#[https://web.archive.org/web/20250808053249/https://news.cornell.edu/stories/2023/11/jurassic-worlds-might-be-easier-spot-modern-earth The Cornell article does not even remotely support the idea that &quot;aliens visited Earth&quot;]. It mentions a ''chance'' of &quot;life there [a habitable exoplanet] might not be limited to microbes, but could include creatures as large and varied as the megalosauruses or microraptors that once roamed Earth.&quot;, but again, no justification to take this article as proof that &quot;aliens may have visited us!&quot;. There's no mention of aliens visiting Earth anywhere in the article. Once again this is only proving the background premise, but not the unsupported, nonsensical &quot;alien likelihood&quot; argument that the author of this garbage page is trying to push so desperately.
:#The Parker Solar Probe WP article does not even mention aliens either. It follows the same issues as the previous argument.
:And the other page this user complained about [https://en.wikiversity.org/wiki/User_talk:Atcovi#Deletion_of_educational_page_because_of_personal_opinion on my talk page] holds almost similar, maybe even more fatal mistakes, than this one. It has nothing to do with &quot;taking offense&quot;, this is just low-quality, garbage content. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:56, 11 April 2026 (UTC)
::Why do you think pro claims are required to be proven? It's possible to object to them and these are arguments, not contextualized to be statements of proven facts. And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future. Instead of insulting others' intelligence, maybe engage with the actual reasoning rather than censoring it away. And there are lots of sources, such as [https://interestingengineering.com/science/alien-civilizations-may-have-visited-earth-millions-of-years-ago-study-says Alien Civilizations May Have Visited Earth Millions of Years Ago, Study Says] etc etc. The sources are used for the arguments themselves individually. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:30, 11 April 2026 (UTC)
:::Because, once again, this is not a site that caters to rampant debating for the sake of &quot;we need to employ rationality and logic to solve the world's problems&quot;, we have policies that we need to fulfill. The claims made in the pro argument clearly do not meet [[Wikiversity:Verifiability]], since you cannot verify these arguments with the sources because they are not relevant. 
:::''&quot;And it's not a strange or unreasonable argument to make that since Earth has existed for long, it's more likely that aliens have come here in the past than in recent times or the near future.&quot;'' The point being is that these arguments are not supported by the sources. Even the article you mention poses the idea as a hypothetical model. This is just you twisting the article to fit your unsupported narrative. I'll bring direct quotes for you to show why the linked article does not help you:
:::* ''One problem the researchers do make sure to point out is that '''they are working with only one data point: our own behaviors and capabilities for space exploration'''. “We tried to come up with a model that would involve the fewest assumptions about sociology that we could,” Carroll-Nellenback told Business Insider. '''We have no real way of knowing the motivations of an alien civilization'''.'' --&gt; proves that this is just speculation and no evidence-based arguments have been provided for the idea that aliens likely visited Earth.
:::And I'm not sure if you read my entire response, but I ''did'' engage with your &quot;actual reasoning&quot; and exposed its weaknesses and lack of adherence to Wikiversity policies. If we allowed content that was just filled with non sequiturs we would have content that fails Wikiversity's educational objectives and reduces the overall quality of this website, hence why such a harsh stance needs to be taken. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:50, 11 April 2026 (UTC)</text>
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&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''Landcare ACT Wellbeing through Nature program evaluation: Final report''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'' (published online ahead of print 2023). https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''Landcare ACT Wellbeing through Nature program evaluation: Final report''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'' (published online ahead of print 2023). https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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      <comment>See also: [[User:Jtneill/Research|Research]]</comment>
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      <text bytes="7465" sha1="o1ogwkca3k2l6fo7iqka4kmqpv6g1hj" xml:space="preserve">See also: [[User:Jtneill/Research|Research]]
==[[../Research/Profiles|Profiles]]==
{{../Research/Profiles}}
&lt;!-- ==2026==
--&gt;
==2025==
{{Hanging indent|1=
Brichacek, A., Neill, J. T., Murray, K., Rieger, E., Watsford, C. (2025). Body Image Flexibility and Inflexibility Scale (BIFIS). In W. Ramseyer Winter, T. L. Tylka, &amp; A. M. Landor (Eds.), ''Handbook of body image-related measures''. Cambridge University Press (pp. 118–121). https://doi.org/10.1017/9781009398275.039

{{User:Jtneill/Publications/2025/Body}}&lt;!--
Neill, J. T., Herbert, S., Hartley, R., &amp; D'Cunha, N. (in preparation). ''Art for Wellbeing at the National Gallery of Australia: Thematic analysis of participant and staff perspectives''.

Lozancic Babic, V. &amp; Neill, J. T. ...  --&gt;
}}

==2024==
{{Hanging indent|1=

Black, H. M., &amp; Neill, J. T. (2024). Wellbeing through nature: A qualitative exploration of psychosocial aspects of a Landcare ACT nature-connection program. ''Journal of Outdoor and Environmental Education''. https://doi.org/10.1007/s42322-024-00184-2

Boerma, M., Beel, N., Neill, J. T., Jeffries, C., Krishnamoorthy, G., &amp; Guerri-Guttenberg, J. (2024). Male-friendly counselling for young men: a thematic analysis of client and caregiver experiences of Menslink counselling. ''Australian Psychologist'', 1–12. https://doi.org/10.1080/00050067.2024.2378119 ([https://www.researchgate.net/publication/385649063_Male-friendly_counselling_for_young_men_a_thematic_analysis_of_client_and_caregiver_experiences_of_Menslink_counselling#fullTextFileContent pdf])

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (accepted). The Body Image Flexibility and Inflexibility Scale (BIFIS). In V. Ramseyer Winter, T.  Tylka, &amp; A. Landor (Eds.), ''Handbook of body image-related measures'' (pp. *–*). Cambridge University Press.

Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2024). The distinct affect regulation functions of body image flexibility and inflexibility: A prospective study in adolescents and emerging adults. ''Body Image'', ''50'', 101726. https://doi.org/10.1016/j.bodyim.2024.101726

{{User:Jtneill/Publications/2024/Collaborative}}

Neill, J. T. &amp; Black. H. (2024). ''Landcare ACT Wellbeing through Nature program evaluation: Final report''. University of Canberra, Australia.

{{User:Jtneill/Publications/2024/Rich}}

}}

==2023==
{{Hanging indent|1=
Brichacek, A. L., Neill, J. T., Murray, K., Rieger, E., &amp; Watsford, C. (2023). Ways of responding to body image threats: Development of the Body Image Flexibility and Inflexibility Scale for Youth. ''Journal of Contextual Behavioral Science'', ''30'', 31–40. https://doi.org/10.1016/j.jcbs.2023.08.007

{{/2023/WIL}}

Ross, B. M., &amp; Neill, J. T. (2023). Exploring the relationship between mental health, drug use, personality, and attitudes towards psilocybin-assisted therapy. ''[https://akjournals.com/view/journals/2054/2054-overview.xml Journal of Psychedelic Studies]'' (published online ahead of print 2023). https://doi.org/10.1556/2054.2023.00264}}

==2022==
{{Hanging indent|1=

Neill, J. T., Goch, I., Sullivan, A., &amp; Simons, M. (2022). The role of burn camp in the recovery of young people from burn injury: A qualitative study using long-term follow-up interviews with parents and participants. ''Burns'', ''48''(5), 1139–1148. https://doi.org/10.1016/j.burns.2021.09.020

Stevenson, D. J., Neill, J. T., Ball, K., Smith, R., &amp; Shores, M. C. (2022). How do preschool to year 6 educators prevent and cope with occupational violence from students? ''Australian Journal of Education'', ''66''(2), 154–170. https://doi.org/10.1177/00049441221092472. [https://www.teachermagazine.com/au_en/articles/the-research-files-episode-77-coping-with-violence-from-students Podcast].

}}

==2021==
{{Hanging indent|1=Brichacek, A. L., Murray, K., Neill, J. T., &amp; Rieger, E. (2021). Contextual behavioral approaches to understanding body image threats and coping in youth: A qualitative study. ''Journal of Adolescent Research'', ''39''(2), 328–360. https://doi.org/10.1177/07435584211007851}}

==2016==
{{Hanging indent|1=

Bowen, D. J., &amp; Neill, J. T. (2016). Effects of the PCYC Catalyst outdoor adventure intervention program on youths' life skills, mental health, and delinquent behaviour. ''International Journal of Adolescence and Youth'', ''21''(1), 34–55. https://doi.org/10.1080/02673843.2015.1027716

{{/2016/Internationalisation}}}}

==2013==
{{Hanging indent|1=
{{/2013/Promoting}}

{{/2013/Teaching}}
}}

==2011==
{{Hanging indent|1=

Gray, T. L. &amp; Neill, J. T. (2011). Program evaluation. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 164–182). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.010

Neill, J. T., &amp; Gray, T. L. (2011). Technology, risk and outdoor programming. In ''[https://www.cambridge.org/core/books/risk-management-in-the-outdoors/8B270918DA02077EB040BF2A4646FA7F Risk management in the outdoors: A whole-of-organisation approach for education, sport and recreation]'' (pp. 132–149). Cambridge University Press. https://doi.org/10.1017/CBO9781139192682.008
}}

==2010==
Mackay, G. J., &amp; Neill, J. T. (2010). The effect of “green exercise” on state anxiety and the role of exercise duration, intensity, and greenness: A quasi-experimental study. ''Psychology of Sport and Exercise'', ''11''(3), 238–245. https://doi.org/10.1016/j.psychsport.2010.01.002

==2008==
{{Hanging indent|1=Neill, J. T. (2008). Enhancing life effectiveness: The impacts of outdoor education programs. [Unpublished doctoral dissertation]. University of Western Sydney. https://researchdirect.westernsydney.edu.au/islandora/object/uws:6441/}}

==2002==
{{/2002/Dramaturgy}}

==1997==
Hattie, J., Marsh, H. W., Neill, J. T., &amp; Richards, G. E. (1997). Adventure education and Outward Bound: Out-of-class experiences that make a lasting difference. ''Review of Educational Research'', ''67''(1), 43-87. https://doi.org/10.3102/00346543067001043

==Reports==
{{Hanging indent|1=
Neill, J. T. &amp; Bowen, D. J. (2014). ''[https://drive.google.com/file/d/0B2N4zSp4hmN9WUF3bzhuZ3JoNGM/view?usp=sharing&amp;resourcekey=0-y0ZTjcdhHXqQKNtz50BW0A Research evaluation of PCYC Bornhoffen Catalyst intervention programs for youth-at-risk &lt;nowiki&gt;[&lt;/nowiki&gt;2012-2013&lt;nowiki&gt;]&lt;/nowiki&gt;]''. University of Canberra.
}}

==Theses==
* [[User:Jtneill/PhD|PhD]]
&lt;!-- 
==Published==
* [http://www.wilderdom.com/JamesNeill/JamesNeillpublications.htm Articles &amp; presentations by James Neill]
--&gt;

==Ideas / In progress==
* [[User:Jtneill/4 pillars of free and open teaching|4 pillars of free and open teaching]]
* Some international trends in outdoor education - Past, present, and future
* Ingando camp (life effectiveness)
* Life Effectiveness Questionnaire psychometrics
* OE outcomes (longitudinal study)
* Adolescent Coping Scale psychometrics
* Resilience Scale psychometrics
* Overview of Outdoor Education Theory and/or Research
* Overview of Outdoor Education in Australia
* Overview of Adventure Therapy Theory and/or Education
* Past Trends and Future Directions for Outdoor Education
* Psychological Aspects of Outdoor Education
* Outdoor Education and Modern Technology
* Outdoor Education and Environmental Sustainability

==See also==
* [[User:Jtneill/Presentations]]</text>
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Sorry! Due to an rendering error, this page cannot be displayed. Try [[Special:Purge|purging]] your browser's memory.&lt;/div&gt;{{daughters}}
{{shortcut|WP}}
{{RoundBoxTop|theme=2}}
These learning resources aim at providing knowledge that all [[w:Wikipedia|Wikipedia]] and [[w:Wikimedia]] users, authors and administrators should possess. It aims at answering questions such as:
* How can wikis and Wikipedia as phenomenon be understood? 
* How to edit and administrate Wikipedia and Mediawiki sites? 
* How can students, teachers, librarians, journalists, etc, use and relate to Wikipedia and other wikis? 
* What are the main criticisms of Wikipedia? 
* What research exists related to Wikipedia as phenomenon?
* What are the current wiki technology development trends? 
{{RoundBoxBottom}}
{{TOCright}}

==About Wikipedia==
:''See more at [[w:Wikipedia|Wikipedia article on Wikipedia]].''
'''Wikipedia''' is a multilingual, [[wiki]]-based, free-content [[wikipedia:Online encyclopedia|online encyclopedia]] project. The name is a portmanteau of the words wiki, the Hawaiian word meaning quick, and encyclopedia. Wikipedia is written collaboratively by volunteers, allowing most of its articles to be edited by almost anyone with access to the website and is a free site for all types of ages. Its main servers are in Tampa, Florida, with additional servers in Amsterdam and Seoul.

Wikipedia was launched as an English language project on January 15, 2001, as a complement to the expert-written and now defunct [[wikipedia:Nupedia|Nupedia]], and is now operated by the non-profit Wikimedia Foundation. It was created by Larry Sanger and Jimmy Wales; Sanger resigned from both Nupedia and Wikipedia on March 1, 2002. Wales has described Wikipedia as &quot;an effort to create and distribute a multilingual free encyclopedia of the highest possible quality to every single person on the planet in their own language&quot;.

Wikipedia has more than fifty eight million articles in many languages, including more than 6 million articles in both the English-language version and the Cebuano-language version and more than two million in the German-language version. There are 250 language editions of Wikipedia, and 18 of them have more than 50,000 articles. The German-language edition has been distributed on DVD-ROM, and there have been proposals for an English DVD or print edition. Since its inception, Wikipedia has steadily risen in popularity, and has spawned several sister projects. According to Alexa, Wikipedia ranks among the top fifteen most visited sites, and many of its pages have been mirrored or forked by other sites, such as Answers.com.

There has been controversy over Wikipedia's reliability and accuracy, with the site receiving criticism for its susceptibility to vandalism, uneven quality and inconsistency, systemic bias, and preference for consensus or popularity over credentials. Information is sometimes unconfirmed and questionable, lacking the proper sources that, in the eyes of most &quot;Wikipedians&quot; (as Wikipedia's contributors call themselves), are necessary for an article to be considered &quot;high quality&quot;. However, a 2005 comparison performed by the science journal Nature of sections of Wikipedia and the Encyclopædia Britannica found that the two were close in terms of the accuracy of their articles on the natural sciences. This study was challenged by Encyclopedia Britannica, Inc., who described it as &quot;fatally flawed&quot;.

==Wikipedia and the Neutrality Principle==
A precis of key ideas from Jose Van Dijck, who wrote a chapter on Wikipedia and the neutrality principle in her 2013 book, The Culture of Connectivity: A Critical History of Social Media (Oxford University Press). 

==Learning resources==
The following is a list of learning resources that may be useful as course material on courses about Wikis and Wikipedia. The list is organized after resource type, and includes resources developed within this project and [[#See also|related Wikiversity projects]] as well as external links.

===Syllabi===
* [[/syllabi|Suggested syllabus]] for a university level course named ''Wikis and Wikipedia - Authoring, Reliability and Technology.''
* [[/Writing an article on Wikipedia]]: a course to take a new user through the process of writing or editing their first article.

===Assignments and exercises===
* [[Wikipedia/Quizzes]] - Multiple choice questions about ''Wikipedia policies'' and ''Wikipedia criticism''.
* [http://www.fanpop.com/spots/wikipedia/quiz/show/3381 Wikipedia pop quiz - at Fanpop.com]  - Multiple choice questions about ''Wikipedia history'' and ''Wikipedia statistics''
* [[Inside_Wikipedia#Discussion]] - Suggested discussion topics about ''Wikipedia policies'' and ''Wikipedia criticism''.
* [[Wikimedia Ethics/Suggested essays]]
* [[Wikipedia/Suggested essays]]
* [[Generating dynamic content with MediaWiki]] - Practical exercises

===Videos ===
These are recorded seminars, lectures, webinars and instructions videos related to Wikipedia:
* [http://webcast.oii.ox.ac.uk/?view=Webcast&amp;ID=20050711_76 Jimmy Wales, &quot;The Intelligence of Wikipedia&quot;, Oxford Internet Institute webcast (54 minutes Real Media or MPEG4)] Wikipedia founder Jimmy Wales explains the history and growth of Wikipedia with a particular focus on the internal community process which ensure a constant growth in quality.
* [http://siliconvalley.wikia.com/wiki/Screencasts Wikia training videos]
* [http://www.youtube.com/results?search_query=wikipedia&amp;search_type=&amp;aq=f Youtube search: Wikipedia]

===Slides===
See also: [[:Commons:Category:Wikimedia_presentations_in_English]] and [[meta:Presentations|meta.wikimedia.org/wiki/Presentations]].

These are slide show presentations, lecture notes and handouts related to Wikipedia:
* [http://library.uncwil.edu/web/instruction/workshops/wikipedia.ppt Anne Pemberton, Rachel Radom, &quot;Wikipedia 360° - The Good, The Bad, and the Anonymous&quot;, Randall Library, University of North Carolina Wilmington, September 2007]
* [http://people.lib.ucdavis.edu/psa/FSUclass.ppt Phoebe S. Ayers, &quot;Wikipedia 101 - or, how I learned to stop worrying and trust the Internet&quot;, UC Davis University Library, California Library Association Annual Meeting, February 2007] 
* [http://web.archive.org/web/20090730203527/http://www.library.ubc.ca/edlib/courses/lled320/Wikiland6.ppt Aubri Keleman, &quot;Teaching in Wikiland - An introduction to using Wikis in the classroom, January 2006] 
* [http://net.educause.edu/ir/library/powerpoint/ELI08216.pps Martha Groom, Andreas Brockhaus, &quot;Using Wikipedia to Re-envision the Term Paper&quot;, University of Washington Bothell, October 2007] 
* [http://trblist.tamu.edu/programs/web_tools/winter.pdf Ken Winter, &quot;Wiki, the Web and WorldCat: Open Editing and Research In Action&quot; Committee on Library and Information Science for Transportation Annual meeting, January 2006]
* [http://www.riehle.org/wp-content/uploads/2008/09/learning-from-wikipedia.pdf Dirk Riehle, &quot;Learning from Wikipedia: Open collaboration within organisations&quot;, Talk the Future - the Future of Work and Business Conference, September 2008]

===Scientific publications===
These are peer-reviewed academic papers related to Wikipedia, that students may review and use as references:
* [[w:Academic studies about Wikipedia|Academic studies about Wikipedia]] A review of some research publications related to Wikipedia
* [[w:Wikipedia:Wikipedia in academic studies|List of academic studies on Wikipedia]] ([[w:WP:ACST]]) List of academic conference presentations, peer-reviewed papers and other types of academic writing which focus on Wikipedia
* [[meta:Wiki Research Bibliography|Wiki Research Bibliography]] - meta list article
* [[:Category:Wikimedia research|Wikimedia research]] - Wikiversity category
* [http://scholar.google.se/scholar?q=wikipedia&amp;hl=sv&amp;lr= Google scholar search: Wikipedia]

===Text books===
These text books may be used as course literature:
* John Broughton, [[w:Wikipedia_-_The_Missing_Manual|''Wikipedia: The Missing Manual'']], O'Reilly, 2008 - [http://books.google.se/books?id=h37N0BvkVSUC&amp;printsec=frontcover Preview]
* {{cite book
|author=[[w:user:phoebe|Phoebe Ayers]]
|author2=[[w:user:Charles Matthews|Charles Matthews]]
|author3=[[w:user:Tlogmer|Ben Yates]]
|title=How Wikipedia Works - And How You Can Be a Part of It
|publisher=[[w:No Starch Press|No Starch Press]]
|location=San Francisco
|date=September 2008
|isbn=978-1-59327-176-3
}}
*  Dan Woods, Peter Thoeny, Ward Cunningham, [[w:Wikis For Dummies|''Wikis For Dummies'']], 2007 - [http://books.google.com/books?id=5VXgXlU7g-YC&amp;printsec=frontcover&amp;hl=sv Preview]
* [[b:Wiki Science|Wiki Science]] - wikibook
* [[b:Starting and Running a Wiki Website|Starting and Running a Wiki Website]] - wikibook 
* [[b:MediaWiki Administrator's Handbook|MediaWiki Administrator's Handbook]] - wikibook
* [[b:MediaWiki User Guide|MediaWiki User Guide]]  - wikibook
* [[w:Wikipedia:Wikipedia in books|List of books which have discussed Wikipedia]]
* [http://books.google.com/books?hl=sv&amp;q=wikipedia Google books search for &quot;Wikipedia&quot;]

===Glossaries===
These glossares may serve as lists of terms that a student should understand after a course about Wikipedia:
* [[w:Wikipedia:Glossary|Wikipedia:Glossary]]
* [[Wikiversity:Glossary]]

===Wikipedia articles===
These are articles, essays and help pages available at en.wikipedia.org, that may be useful as course materials on a course about Wikipedia:
* [[w:Wikipedia:Introduction|Introduction]]
* [[w:Welcoming committee/Welcome to Wikipedia|Welcome to Wikipedia]]
* [[w:Help:Contents|Help pages]]
* [[w:Criticism of Wikipedia|Criticism of Wikipedia]]
* [[w:Reliability of Wikipedia|Reliability of Wikipedia]]
* [[w:Wikipedia:Citing Wikipedia|Citing Wikipedia]]
* [[Researching with Wikipedia]]
* [[w:History of Wikipedia|History of Wikipedia]]
* [[w:Wiki|Wiki]]
* [[w:Wikipedia|Wikipedia]]
* [[w:Wikipedia:Criticisms|Criticisms]]
* [[w:Wikipedia:Why Wikipedia is not so great|Why Wikipedia is not so great]]
* [[w:Wikipedia:School and university projects|School and university projects]]

===Case studies===
* [[Wikimedia Ethics/Case Studies|Wikimedia Ethics Case Studies]]

===Statistical sources===
* [[w:Wikipedia:Statistics|Wikipedia statistics]] 
* [http://stats.grok.se/ Wikipedia article traffic statistics]  - Comparison of invidual articles
* [http://stats.wikimedia.org/ Wikimedia official statistics] - Comparison of wiki language versions
* [http://s23.org/wikistats/ Wikistats by S23 - Statistics about Mediawikis] - Comparison of wiki language versions
* [[meta:List_of_Wikipedias|List of Wikipedias]] - Comparison of wiki language versions
* [[meta:List_of_Wikimedians_by_religion#Percentages.2C_world_versus_Wikimedians.|Poll of Wikimedians by religion]]

===University level courses===
* [[:wikipedia:Wikipedia:School_and_university_projects#Mid_Sweden_University_course_about_wikis_and_Wikipedia_.28Fall_2009_and_Summer_2010.29|Informatics BA (A), Wikipedia – Authoring, Reliability and Technology, 7.5 higher education credits]], Swedish distance course offered from fall 2009.

===Other courses, workshops, conferences and events===
* [[w:Wikipedia:Academy|Wikipedia academies]]
* [[w:Wikimania|Wikimania]]
* Elective part of the Australian [[w:Higher School Certificate|Higher School Certificate]] syllabus [http://web.archive.org/web/20070902171026/http://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/english-prescription-09-12.doc English stage 6], Standard Module C, Elective 1: The Global Village, 2009-2012 (Board of Studies, New South Wales). See media coverage: [[w:Wikipedia:Wikipedia_Signpost/2008-05-26/In_the_news|Wikipedia Signpost]], [http://www.brisbanetimes.com.au/articles/2008/05/26/1211653992226.html www.brisbanetimes.com].

===Other lists of learning resources===
* [http://www.det.wa.edu.au/education/cmis/eval/curriculum/ict/wikis/ Government of Western Austarialian, Department of education &amp; training, Resourcing the Curriculum: Wikis in te classroom]
* [http://www.det.wa.edu.au/education/cmis/eval/curriculum/ict/wikipedia/ Government of Western Austarialian, Department of education &amp; training, Resourcing the Curriculum: Wikipedia]
* [http://teachwtech.blogspot.com/2006/03/k-12-wiki-resources.html K-12 wiki resources]
* [http://ocw.usu.edu/Instructional_Technology/new-media/wikis.htm#Whats_a_Wiki.3F Utah State University list of resources: Wiki]
* [http://ocw.usu.edu/Instructional_Technology/principles-and-practices-of-technology/web-2.0-wikis/?searchterm=wikis Utah State University list of resources: Web2.0: Wikis]

==See also==
See also other wikiversity resources and pages:
===Wiki===
* [[Learn to learn a wiki way]] 
* [[Learning to edit a wiki]]
* [[Wiki]]
* [[Portal:Wiki|Wiki portal]]

===Wikipedia===
{{wikipedia}}
* [[Inside Wikipedia]]
* [[Pywikipediabot]]
* [[Wikipedia service-learning courses]]
* [[Is Wikipedia a legitimate research source?]]

===MediaWiki===
* [[Generating dynamic content with MediaWiki]]  
* [[MediaWiki Project]]
* [[Topic:MediaWiki|MediaWiki]] - a [[Wikiversity:Topics|Wikiversity topic]]

===Wikimedia===
* [[How to be a Wikimedia sysop]]
* [[Portal:Wikimedia Studies|Wikimedia studies portal]] - a [[Wikiversity:Portals|Wikiversity portal]]
* [[School:WikiService|School of WikiService]]
* [[Wikimedia Ethics]]

===Wikipedia related===
* [http://www.wikigame.org WikiGame]

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      <text bytes="13260" sha1="2f9jgfnh9j0smk5npb6w3atuxgr22lg" xml:space="preserve">{{daughters}}
{{shortcut|WP}}
{{RoundBoxTop|theme=2}}
These learning resources aim at providing knowledge that all [[w:Wikipedia|Wikipedia]] and [[w:Wikimedia]] users, authors and administrators should possess. It aims at answering questions such as:
* How can wikis and Wikipedia as phenomenon be understood? 
* How to edit and administrate Wikipedia and Mediawiki sites? 
* How can students, teachers, librarians, journalists, etc, use and relate to Wikipedia and other wikis? 
* What are the main criticisms of Wikipedia? 
* What research exists related to Wikipedia as phenomenon?
* What are the current wiki technology development trends? 
{{RoundBoxBottom}}
{{TOCright}}

==About Wikipedia==
:''See more at [[w:Wikipedia|Wikipedia article on Wikipedia]].''
'''Wikipedia''' is a multilingual, [[wiki]]-based, free-content [[wikipedia:Online encyclopedia|online encyclopedia]] project. The name is a portmanteau of the words wiki, the Hawaiian word meaning quick, and encyclopedia. Wikipedia is written collaboratively by volunteers, allowing most of its articles to be edited by almost anyone with access to the website and is a free site for all types of ages. Its main servers are in Tampa, Florida, with additional servers in Amsterdam and Seoul.

Wikipedia was launched as an English language project on January 15, 2001, as a complement to the expert-written and now defunct [[wikipedia:Nupedia|Nupedia]], and is now operated by the non-profit Wikimedia Foundation. It was created by Larry Sanger and Jimmy Wales; Sanger resigned from both Nupedia and Wikipedia on March 1, 2002. Wales has described Wikipedia as &quot;an effort to create and distribute a multilingual free encyclopedia of the highest possible quality to every single person on the planet in their own language&quot;.

Wikipedia has more than fifty eight million articles in many languages, including more than 6 million articles in both the English-language version and the Cebuano-language version and more than two million in the German-language version. There are 250 language editions of Wikipedia, and 18 of them have more than 50,000 articles. The German-language edition has been distributed on DVD-ROM, and there have been proposals for an English DVD or print edition. Since its inception, Wikipedia has steadily risen in popularity, and has spawned several sister projects. According to Alexa, Wikipedia ranks among the top fifteen most visited sites, and many of its pages have been mirrored or forked by other sites, such as Answers.com.

There has been controversy over Wikipedia's reliability and accuracy, with the site receiving criticism for its susceptibility to vandalism, uneven quality and inconsistency, systemic bias, and preference for consensus or popularity over credentials. Information is sometimes unconfirmed and questionable, lacking the proper sources that, in the eyes of most &quot;Wikipedians&quot; (as Wikipedia's contributors call themselves), are necessary for an article to be considered &quot;high quality&quot;. However, a 2005 comparison performed by the science journal Nature of sections of Wikipedia and the Encyclopædia Britannica found that the two were close in terms of the accuracy of their articles on the natural sciences. This study was challenged by Encyclopedia Britannica, Inc., who described it as &quot;fatally flawed&quot;.

==Wikipedia and the Neutrality Principle==
A precis of key ideas from Jose Van Dijck, who wrote a chapter on Wikipedia and the neutrality principle in her 2013 book, The Culture of Connectivity: A Critical History of Social Media (Oxford University Press). 

==Learning resources==
The following is a list of learning resources that may be useful as course material on courses about Wikis and Wikipedia. The list is organized after resource type, and includes resources developed within this project and [[#See also|related Wikiversity projects]] as well as external links.

===Syllabi===
* [[/syllabi|Suggested syllabus]] for a university level course named ''Wikis and Wikipedia - Authoring, Reliability and Technology.''
* [[/Writing an article on Wikipedia]]: a course to take a new user through the process of writing or editing their first article.

===Assignments and exercises===
* [[Wikipedia/Quizzes]] - Multiple choice questions about ''Wikipedia policies'' and ''Wikipedia criticism''.
* [http://www.fanpop.com/spots/wikipedia/quiz/show/3381 Wikipedia pop quiz - at Fanpop.com]  - Multiple choice questions about ''Wikipedia history'' and ''Wikipedia statistics''
* [[Inside_Wikipedia#Discussion]] - Suggested discussion topics about ''Wikipedia policies'' and ''Wikipedia criticism''.
* [[Wikimedia Ethics/Suggested essays]]
* [[Wikipedia/Suggested essays]]
* [[Generating dynamic content with MediaWiki]] - Practical exercises

===Videos ===
These are recorded seminars, lectures, webinars and instructions videos related to Wikipedia:
* [http://webcast.oii.ox.ac.uk/?view=Webcast&amp;ID=20050711_76 Jimmy Wales, &quot;The Intelligence of Wikipedia&quot;, Oxford Internet Institute webcast (54 minutes Real Media or MPEG4)] Wikipedia founder Jimmy Wales explains the history and growth of Wikipedia with a particular focus on the internal community process which ensure a constant growth in quality.
* [http://siliconvalley.wikia.com/wiki/Screencasts Wikia training videos]
* [http://www.youtube.com/results?search_query=wikipedia&amp;search_type=&amp;aq=f Youtube search: Wikipedia]

===Slides===
See also: [[:Commons:Category:Wikimedia_presentations_in_English]] and [[meta:Presentations|meta.wikimedia.org/wiki/Presentations]].

These are slide show presentations, lecture notes and handouts related to Wikipedia:
* [http://library.uncwil.edu/web/instruction/workshops/wikipedia.ppt Anne Pemberton, Rachel Radom, &quot;Wikipedia 360° - The Good, The Bad, and the Anonymous&quot;, Randall Library, University of North Carolina Wilmington, September 2007]
* [http://people.lib.ucdavis.edu/psa/FSUclass.ppt Phoebe S. Ayers, &quot;Wikipedia 101 - or, how I learned to stop worrying and trust the Internet&quot;, UC Davis University Library, California Library Association Annual Meeting, February 2007] 
* [http://web.archive.org/web/20090730203527/http://www.library.ubc.ca/edlib/courses/lled320/Wikiland6.ppt Aubri Keleman, &quot;Teaching in Wikiland - An introduction to using Wikis in the classroom, January 2006] 
* [http://net.educause.edu/ir/library/powerpoint/ELI08216.pps Martha Groom, Andreas Brockhaus, &quot;Using Wikipedia to Re-envision the Term Paper&quot;, University of Washington Bothell, October 2007] 
* [http://trblist.tamu.edu/programs/web_tools/winter.pdf Ken Winter, &quot;Wiki, the Web and WorldCat: Open Editing and Research In Action&quot; Committee on Library and Information Science for Transportation Annual meeting, January 2006]
* [http://www.riehle.org/wp-content/uploads/2008/09/learning-from-wikipedia.pdf Dirk Riehle, &quot;Learning from Wikipedia: Open collaboration within organisations&quot;, Talk the Future - the Future of Work and Business Conference, September 2008]

===Scientific publications===
These are peer-reviewed academic papers related to Wikipedia, that students may review and use as references:
* [[w:Academic studies about Wikipedia|Academic studies about Wikipedia]] A review of some research publications related to Wikipedia
* [[w:Wikipedia:Wikipedia in academic studies|List of academic studies on Wikipedia]] ([[w:WP:ACST]]) List of academic conference presentations, peer-reviewed papers and other types of academic writing which focus on Wikipedia
* [[meta:Wiki Research Bibliography|Wiki Research Bibliography]] - meta list article
* [[:Category:Wikimedia research|Wikimedia research]] - Wikiversity category
* [http://scholar.google.se/scholar?q=wikipedia&amp;hl=sv&amp;lr= Google scholar search: Wikipedia]

===Text books===
These text books may be used as course literature:
* John Broughton, [[w:Wikipedia_-_The_Missing_Manual|''Wikipedia: The Missing Manual'']], O'Reilly, 2008 - [http://books.google.se/books?id=h37N0BvkVSUC&amp;printsec=frontcover Preview]
* {{cite book
|author=[[w:user:phoebe|Phoebe Ayers]]
|author2=[[w:user:Charles Matthews|Charles Matthews]]
|author3=[[w:user:Tlogmer|Ben Yates]]
|title=How Wikipedia Works - And How You Can Be a Part of It
|publisher=[[w:No Starch Press|No Starch Press]]
|location=San Francisco
|date=September 2008
|isbn=978-1-59327-176-3
}}
*  Dan Woods, Peter Thoeny, Ward Cunningham, [[w:Wikis For Dummies|''Wikis For Dummies'']], 2007 - [http://books.google.com/books?id=5VXgXlU7g-YC&amp;printsec=frontcover&amp;hl=sv Preview]
* [[b:Wiki Science|Wiki Science]] - wikibook
* [[b:Starting and Running a Wiki Website|Starting and Running a Wiki Website]] - wikibook 
* [[b:MediaWiki Administrator's Handbook|MediaWiki Administrator's Handbook]] - wikibook
* [[b:MediaWiki User Guide|MediaWiki User Guide]]  - wikibook
* [[w:Wikipedia:Wikipedia in books|List of books which have discussed Wikipedia]]
* [http://books.google.com/books?hl=sv&amp;q=wikipedia Google books search for &quot;Wikipedia&quot;]

===Glossaries===
These glossares may serve as lists of terms that a student should understand after a course about Wikipedia:
* [[w:Wikipedia:Glossary|Wikipedia:Glossary]]
* [[Wikiversity:Glossary]]

===Wikipedia articles===
These are articles, essays and help pages available at en.wikipedia.org, that may be useful as course materials on a course about Wikipedia:
* [[w:Wikipedia:Introduction|Introduction]]
* [[w:Welcoming committee/Welcome to Wikipedia|Welcome to Wikipedia]]
* [[w:Help:Contents|Help pages]]
* [[w:Criticism of Wikipedia|Criticism of Wikipedia]]
* [[w:Reliability of Wikipedia|Reliability of Wikipedia]]
* [[w:Wikipedia:Citing Wikipedia|Citing Wikipedia]]
* [[Researching with Wikipedia]]
* [[w:History of Wikipedia|History of Wikipedia]]
* [[w:Wiki|Wiki]]
* [[w:Wikipedia|Wikipedia]]
* [[w:Wikipedia:Criticisms|Criticisms]]
* [[w:Wikipedia:Why Wikipedia is not so great|Why Wikipedia is not so great]]
* [[w:Wikipedia:School and university projects|School and university projects]]

===Case studies===
* [[Wikimedia Ethics/Case Studies|Wikimedia Ethics Case Studies]]

===Statistical sources===
* [[w:Wikipedia:Statistics|Wikipedia statistics]] 
* [http://stats.grok.se/ Wikipedia article traffic statistics]  - Comparison of invidual articles
* [http://stats.wikimedia.org/ Wikimedia official statistics] - Comparison of wiki language versions
* [http://s23.org/wikistats/ Wikistats by S23 - Statistics about Mediawikis] - Comparison of wiki language versions
* [[meta:List_of_Wikipedias|List of Wikipedias]] - Comparison of wiki language versions
* [[meta:List_of_Wikimedians_by_religion#Percentages.2C_world_versus_Wikimedians.|Poll of Wikimedians by religion]]

===University level courses===
* [[:wikipedia:Wikipedia:School_and_university_projects#Mid_Sweden_University_course_about_wikis_and_Wikipedia_.28Fall_2009_and_Summer_2010.29|Informatics BA (A), Wikipedia – Authoring, Reliability and Technology, 7.5 higher education credits]], Swedish distance course offered from fall 2009.

===Other courses, workshops, conferences and events===
* [[w:Wikipedia:Academy|Wikipedia academies]]
* [[w:Wikimania|Wikimania]]
* Elective part of the Australian [[w:Higher School Certificate|Higher School Certificate]] syllabus [http://web.archive.org/web/20070902171026/http://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/english-prescription-09-12.doc English stage 6], Standard Module C, Elective 1: The Global Village, 2009-2012 (Board of Studies, New South Wales). See media coverage: [[w:Wikipedia:Wikipedia_Signpost/2008-05-26/In_the_news|Wikipedia Signpost]], [http://www.brisbanetimes.com.au/articles/2008/05/26/1211653992226.html www.brisbanetimes.com].

===Other lists of learning resources===
* [http://www.det.wa.edu.au/education/cmis/eval/curriculum/ict/wikis/ Government of Western Austarialian, Department of education &amp; training, Resourcing the Curriculum: Wikis in te classroom]
* [http://www.det.wa.edu.au/education/cmis/eval/curriculum/ict/wikipedia/ Government of Western Austarialian, Department of education &amp; training, Resourcing the Curriculum: Wikipedia]
* [http://teachwtech.blogspot.com/2006/03/k-12-wiki-resources.html K-12 wiki resources]
* [http://ocw.usu.edu/Instructional_Technology/new-media/wikis.htm#Whats_a_Wiki.3F Utah State University list of resources: Wiki]
* [http://ocw.usu.edu/Instructional_Technology/principles-and-practices-of-technology/web-2.0-wikis/?searchterm=wikis Utah State University list of resources: Web2.0: Wikis]

==See also==
See also other wikiversity resources and pages:
===Wiki===
* [[Learn to learn a wiki way]] 
* [[Learning to edit a wiki]]
* [[Wiki]]
* [[Portal:Wiki|Wiki portal]]

===Wikipedia===
{{wikipedia}}
* [[Inside Wikipedia]]
* [[Pywikipediabot]]
* [[Wikipedia service-learning courses]]
* [[Is Wikipedia a legitimate research source?]]

===MediaWiki===
* [[Generating dynamic content with MediaWiki]]  
* [[MediaWiki Project]]
* [[Topic:MediaWiki|MediaWiki]] - a [[Wikiversity:Topics|Wikiversity topic]]

===Wikimedia===
* [[How to be a Wikimedia sysop]]
* [[Portal:Wikimedia Studies|Wikimedia studies portal]] - a [[Wikiversity:Portals|Wikiversity portal]]
* [[School:WikiService|School of WikiService]]
* [[Wikimedia Ethics]]

===Wikipedia related===
* [http://www.wikigame.org WikiGame]

[[Category:Wikipedia]]</text>
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See also: [[User:Jtneill/Publications|Publications]]

==Profiles==
* [https://scholar.google.com/citations?user=A5JbI2QAAAAJ&amp;hl=en&amp;oi=ao Google Scholar]
* [https://orcid.org/0000-0003-0710-4550 ORCID]
* [http://www.canberra.edu.au/about-uc/faculties/health/courses/psychology/tabs/staff-profiles/staff-profiles/academic-staff/neill-james Pure] (University of Canberra)
&lt;!-- {{note|Material from [http://ucspace.canberra.edu.au/display/~s613374/Research research] (ucspace) to be transferred here.}} --&gt;

==Projects==
# [[Adventure therapy]]
# [[Collaboration, open research and the commons]]
# [[User:Jtneill/Research/Coping effectiveness|Coping effectiveness]]
# [[User:Jtneill/Research/Fitspiration|Fitspiration]]
# [[Flexible delivery methods and achievement of learning outcomes in psychology]]
# [[Green exercise]] - What are the stress and mood effects of green exercise? To what extent are these effects attributable to individual psychological-, activity-, or environment-related variables?
&lt;!-- # Green exercise journal article with 2010 Hons student – revise draft &amp; return to coauthor --&gt;
# [[Internationalisation of curriculum]] - ALTC IaH research projects
# [[User:Jtneill/Research/Internationalisation of online learning resources|Internationalisation of online learning resources]]
# Life effectiveness
## [[User:Jtneill/Research/Life effectiveness and outdoor education|Life effectiveness and outdoor education]]
## Life effectiveness questionnaire
# [[User:Jtneill/Research/Mess and neatness|Mess and neatness]]
# [[Open academia]] - Academics' motivations, attitudes and behaviours towards openness of access, licensing, formats for research, teaching and service
# [[Open academic practice and Excellence in Research Australia]] - Review of ERA discipline journal list with regard to openness criteria with approx. 6 other UC academics
# Openness of ERA psychology journal review – data collection for A* journals
# [[Outdoor education]] - Theory, processes and outcomes including [[life effectiveness]], health and well-being (Health and well-being impacts of Outward Bound 9-day expedition programs with adolescents)
## Outdoor education and development of a growth mindset
&lt;!-- # Outdoor education journal article with 2010 Hons student – organise data entry for new data --&gt;
# [[Psychology of nature scenes]] - What are the mood effects of digitally presented natural vs. urban scenes?
# [[Psychology of online social networking]] - What motivates engagement in OSN behaviour? What are the associated thoughts and feelings about OSN behaviour?&lt;noinclude&gt;

==See also==
# [[../Presentations|Presentations]]
# [[/Publications|Publications]]
# [[/Questions|Questions]]
# [[/Grants|Grants]]
# [[/Supervision|Supervision]]
# [[/Reviewing|Reviewing]]
# [[User:Jtneill/Teaching]]
# [[Free online peer reviewed journals]]
# [[Open science]]
# [[Original research]]
# [[Portal:Wiki Scholar]]
# [[Wikiversity:Research]]
&lt;!--
==External links==
# [http://wilderdom.info/wiki/User:James_Neill/Publications James Neill's publications]
--&gt;
[[Category:User:Jtneill/Research| ]]&lt;/noinclude&gt;</text>
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See also: [[User:Jtneill/Publications|Publications]]

==Profiles==
* [https://scholar.google.com/citations?user=A5JbI2QAAAAJ&amp;hl=en&amp;oi=ao Google Scholar]
* [https://orcid.org/0000-0003-0710-4550 ORCID]
* [http://www.canberra.edu.au/about-uc/faculties/health/courses/psychology/tabs/staff-profiles/staff-profiles/academic-staff/neill-james Pure] (University of Canberra)
&lt;!-- {{note|Material from [http://ucspace.canberra.edu.au/display/~s613374/Research research] (ucspace) to be transferred here.}} --&gt;

==Projects==
# [[Adventure therapy]]
# [[Collaboration, open research and the commons]]
# [[User:Jtneill/Research/Coping effectiveness|Coping effectiveness]]
# [[User:Jtneill/Research/Fitspiration|Fitspiration]]
# [[Flexible delivery methods and achievement of learning outcomes in psychology]]
# [[Green exercise]] - What are the stress and mood effects of green exercise? To what extent are these effects attributable to individual psychological-, activity-, or environment-related variables?
&lt;!-- # Green exercise journal article with 2010 Hons student – revise draft &amp; return to coauthor --&gt;
# [[Internationalisation of curriculum]] - ALTC IaH research projects
# [[User:Jtneill/Research/Internationalisation of online learning resources|Internationalisation of online learning resources]]
# Life effectiveness
## [[User:Jtneill/Research/Life effectiveness and outdoor education|Life effectiveness and outdoor education]]
## Life effectiveness questionnaire
# [[User:Jtneill/Research/Mess and neatness|Mess and neatness]]
# [[Open academia]] - Academics' motivations, attitudes and behaviours towards openness of access, licensing, formats for research, teaching and service
# [[Open academic practice and Excellence in Research Australia]] - Review of ERA discipline journal list with regard to openness criteria with approx. 6 other UC academics
# Openness of ERA psychology journal review – data collection for A* journals
# [[Outdoor education]] - Theory, processes and outcomes including [[life effectiveness]], health and well-being (Health and well-being impacts of Outward Bound 9-day expedition programs with adolescents)
## Outdoor education and development of a growth mindset
&lt;!-- # Outdoor education journal article with 2010 Hons student – organise data entry for new data --&gt;
# [[Psychology of nature scenes]] - What are the mood effects of digitally presented natural vs. urban scenes?
# [[Psychology of online social networking]] - What motivates engagement in OSN behaviour? What are the associated thoughts and feelings about OSN behaviour?&lt;noinclude&gt;

==See also==
# [[../Presentations|Presentations]]
# [[../Publications|Publications]]
# [[/Questions|Questions]]
# [[/Grants|Grants]]
# [[/Supervision|Supervision]]
# [[/Reviewing|Reviewing]]
# [[User:Jtneill/Teaching]]
# [[Free online peer reviewed journals]]
# [[Open science]]
# [[Original research]]
# [[Portal:Wiki Scholar]]
# [[Wikiversity:Research]]
&lt;!--
==External links==
# [http://wilderdom.info/wiki/User:James_Neill/Publications James Neill's publications]
--&gt;
[[Category:User:Jtneill/Research| ]]&lt;/noinclude&gt;</text>
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&lt;/noinclude&gt;
See also: [[User:Jtneill/Publications|Publications]]

==[[/Profiles|Profiles]]==
{{/Profiles}}
==Projects==
# [[Adventure therapy]]
# [[Collaboration, open research and the commons]]
# [[User:Jtneill/Research/Coping effectiveness|Coping effectiveness]]
# [[User:Jtneill/Research/Fitspiration|Fitspiration]]
# [[Flexible delivery methods and achievement of learning outcomes in psychology]]
# [[Green exercise]] - What are the stress and mood effects of green exercise? To what extent are these effects attributable to individual psychological-, activity-, or environment-related variables?
&lt;!-- # Green exercise journal article with 2010 Hons student – revise draft &amp; return to coauthor --&gt;
# [[Internationalisation of curriculum]] - ALTC IaH research projects
# [[User:Jtneill/Research/Internationalisation of online learning resources|Internationalisation of online learning resources]]
# Life effectiveness
## [[User:Jtneill/Research/Life effectiveness and outdoor education|Life effectiveness and outdoor education]]
## Life effectiveness questionnaire
# [[User:Jtneill/Research/Mess and neatness|Mess and neatness]]
# [[Open academia]] - Academics' motivations, attitudes and behaviours towards openness of access, licensing, formats for research, teaching and service
# [[Open academic practice and Excellence in Research Australia]] - Review of ERA discipline journal list with regard to openness criteria with approx. 6 other UC academics
# Openness of ERA psychology journal review – data collection for A* journals
# [[Outdoor education]] - Theory, processes and outcomes including [[life effectiveness]], health and well-being (Health and well-being impacts of Outward Bound 9-day expedition programs with adolescents)
## Outdoor education and development of a growth mindset
&lt;!-- # Outdoor education journal article with 2010 Hons student – organise data entry for new data --&gt;
# [[Psychology of nature scenes]] - What are the mood effects of digitally presented natural vs. urban scenes?
# [[Psychology of online social networking]] - What motivates engagement in OSN behaviour? What are the associated thoughts and feelings about OSN behaviour?&lt;noinclude&gt;

==See also==
# [[../Presentations|Presentations]]
# [[../Publications|Publications]]
# [[/Questions|Questions]]
# [[/Grants|Grants]]
# [[/Supervision|Supervision]]
# [[/Reviewing|Reviewing]]
# [[User:Jtneill/Teaching]]
# [[Free online peer reviewed journals]]
# [[Open science]]
# [[Original research]]
# [[Portal:Wiki Scholar]]
# [[Wikiversity:Research]]
&lt;!--
==External links==
# [http://wilderdom.info/wiki/User:James_Neill/Publications James Neill's publications]
--&gt;
[[Category:User:Jtneill/Research| ]]&lt;/noinclude&gt;</text>
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See also: [[User:Jtneill/Publications|Publications]]

&lt;include&gt;[[/Research|My research]] seeks to understand the psychosocial processes and effects of experiential, challenge-based programs such as [[outdoor education]], [[adventure therapy]], and [[green exercise]]. More broadly, I am interested in intersections of [[positive psychology]] and [[environmental psychology]].&lt;/include&gt;&lt;noinclude&gt;
==[[/Profiles|Profiles]]==
{{/Profiles}}
==Projects==
# [[Adventure therapy]]
# [[Collaboration, open research and the commons]]
# [[User:Jtneill/Research/Coping effectiveness|Coping effectiveness]]
# [[User:Jtneill/Research/Fitspiration|Fitspiration]]
# [[Flexible delivery methods and achievement of learning outcomes in psychology]]
# [[Green exercise]] - What are the stress and mood effects of green exercise? To what extent are these effects attributable to individual psychological-, activity-, or environment-related variables?
&lt;!-- # Green exercise journal article with 2010 Hons student – revise draft &amp; return to coauthor --&gt;
# [[Internationalisation of curriculum]] - ALTC IaH research projects
# [[User:Jtneill/Research/Internationalisation of online learning resources|Internationalisation of online learning resources]]
# Life effectiveness
## [[User:Jtneill/Research/Life effectiveness and outdoor education|Life effectiveness and outdoor education]]
## Life effectiveness questionnaire
# [[User:Jtneill/Research/Mess and neatness|Mess and neatness]]
# [[Open academia]] - Academics' motivations, attitudes and behaviours towards openness of access, licensing, formats for research, teaching and service
# [[Open academic practice and Excellence in Research Australia]] - Review of ERA discipline journal list with regard to openness criteria with approx. 6 other UC academics
# Openness of ERA psychology journal review – data collection for A* journals
# [[Outdoor education]] - Theory, processes and outcomes including [[life effectiveness]], health and well-being (Health and well-being impacts of Outward Bound 9-day expedition programs with adolescents)
## Outdoor education and development of a growth mindset
&lt;!-- # Outdoor education journal article with 2010 Hons student – organise data entry for new data --&gt;
# [[Psychology of nature scenes]] - What are the mood effects of digitally presented natural vs. urban scenes?
# [[Psychology of online social networking]] - What motivates engagement in OSN behaviour? What are the associated thoughts and feelings about OSN behaviour?&lt;noinclude&gt;

==See also==
# [[../Presentations|Presentations]]
# [[../Publications|Publications]]
# [[/Questions|Questions]]
# [[/Grants|Grants]]
# [[/Supervision|Supervision]]
# [[/Reviewing|Reviewing]]
# [[User:Jtneill/Teaching]]
# [[Free online peer reviewed journals]]
# [[Open science]]
# [[Original research]]
# [[Portal:Wiki Scholar]]
# [[Wikiversity:Research]]
&lt;!--
==External links==
# [http://wilderdom.info/wiki/User:James_Neill/Publications James Neill's publications]
--&gt;
[[Category:User:Jtneill/Research| ]]&lt;/noinclude&gt;</text>
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See also: [[User:Jtneill/Publications|Publications]]

&lt;/noinclude&gt;[[/Research|My research]] seeks to understand the psychosocial processes and effects of experiential, challenge-based programs such as [[outdoor education]], [[adventure therapy]], and [[green exercise]]. More broadly, I am interested in intersections of [[positive psychology]] and [[environmental psychology]].&lt;noinclude&gt;
==[[/Profiles|Profiles]]==
{{/Profiles}}
==Projects==
# [[Adventure therapy]]
# [[Collaboration, open research and the commons]]
# [[User:Jtneill/Research/Coping effectiveness|Coping effectiveness]]
# [[User:Jtneill/Research/Fitspiration|Fitspiration]]
# [[Flexible delivery methods and achievement of learning outcomes in psychology]]
# [[Green exercise]] - What are the stress and mood effects of green exercise? To what extent are these effects attributable to individual psychological-, activity-, or environment-related variables?
&lt;!-- # Green exercise journal article with 2010 Hons student – revise draft &amp; return to coauthor --&gt;
# [[Internationalisation of curriculum]] - ALTC IaH research projects
# [[User:Jtneill/Research/Internationalisation of online learning resources|Internationalisation of online learning resources]]
# Life effectiveness
## [[User:Jtneill/Research/Life effectiveness and outdoor education|Life effectiveness and outdoor education]]
## Life effectiveness questionnaire
# [[User:Jtneill/Research/Mess and neatness|Mess and neatness]]
# [[Open academia]] - Academics' motivations, attitudes and behaviours towards openness of access, licensing, formats for research, teaching and service
# [[Open academic practice and Excellence in Research Australia]] - Review of ERA discipline journal list with regard to openness criteria with approx. 6 other UC academics
# Openness of ERA psychology journal review – data collection for A* journals
# [[Outdoor education]] - Theory, processes and outcomes including [[life effectiveness]], health and well-being (Health and well-being impacts of Outward Bound 9-day expedition programs with adolescents)
## Outdoor education and development of a growth mindset
&lt;!-- # Outdoor education journal article with 2010 Hons student – organise data entry for new data --&gt;
# [[Psychology of nature scenes]] - What are the mood effects of digitally presented natural vs. urban scenes?
# [[Psychology of online social networking]] - What motivates engagement in OSN behaviour? What are the associated thoughts and feelings about OSN behaviour?&lt;noinclude&gt;

==See also==
# [[../Presentations|Presentations]]
# [[../Publications|Publications]]
# [[/Questions|Questions]]
# [[/Grants|Grants]]
# [[/Supervision|Supervision]]
# [[/Reviewing|Reviewing]]
# [[User:Jtneill/Teaching]]
# [[Free online peer reviewed journals]]
# [[Open science]]
# [[Original research]]
# [[Portal:Wiki Scholar]]
# [[Wikiversity:Research]]
&lt;!--
==External links==
# [http://wilderdom.info/wiki/User:James_Neill/Publications James Neill's publications]
--&gt;
[[Category:User:Jtneill/Research| ]]&lt;/noinclude&gt;</text>
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See also: [[User:Jtneill/Publications|Publications]]

&lt;/noinclude&gt;[[User:Jtneill/Research|My research]] seeks to understand the psychosocial processes and effects of experiential, challenge-based programs such as [[outdoor education]], [[adventure therapy]], and [[green exercise]]. More broadly, I am interested in intersections of [[positive psychology]] and [[environmental psychology]].&lt;noinclude&gt;
==[[/Profiles|Profiles]]==
{{/Profiles}}
==Projects==
# [[Adventure therapy]]
# [[Collaboration, open research and the commons]]
# [[User:Jtneill/Research/Coping effectiveness|Coping effectiveness]]
# [[User:Jtneill/Research/Fitspiration|Fitspiration]]
# [[Flexible delivery methods and achievement of learning outcomes in psychology]]
# [[Green exercise]] - What are the stress and mood effects of green exercise? To what extent are these effects attributable to individual psychological-, activity-, or environment-related variables?
&lt;!-- # Green exercise journal article with 2010 Hons student – revise draft &amp; return to coauthor --&gt;
# [[Internationalisation of curriculum]] - ALTC IaH research projects
# [[User:Jtneill/Research/Internationalisation of online learning resources|Internationalisation of online learning resources]]
# Life effectiveness
## [[User:Jtneill/Research/Life effectiveness and outdoor education|Life effectiveness and outdoor education]]
## Life effectiveness questionnaire
# [[User:Jtneill/Research/Mess and neatness|Mess and neatness]]
# [[Open academia]] - Academics' motivations, attitudes and behaviours towards openness of access, licensing, formats for research, teaching and service
# [[Open academic practice and Excellence in Research Australia]] - Review of ERA discipline journal list with regard to openness criteria with approx. 6 other UC academics
# Openness of ERA psychology journal review – data collection for A* journals
# [[Outdoor education]] - Theory, processes and outcomes including [[life effectiveness]], health and well-being (Health and well-being impacts of Outward Bound 9-day expedition programs with adolescents)
## Outdoor education and development of a growth mindset
&lt;!-- # Outdoor education journal article with 2010 Hons student – organise data entry for new data --&gt;
# [[Psychology of nature scenes]] - What are the mood effects of digitally presented natural vs. urban scenes?
# [[Psychology of online social networking]] - What motivates engagement in OSN behaviour? What are the associated thoughts and feelings about OSN behaviour?&lt;noinclude&gt;

==See also==
# [[../Presentations|Presentations]]
# [[../Publications|Publications]]
# [[/Questions|Questions]]
# [[/Grants|Grants]]
# [[/Supervision|Supervision]]
# [[/Reviewing|Reviewing]]
# [[User:Jtneill/Teaching]]
# [[Free online peer reviewed journals]]
# [[Open science]]
# [[Original research]]
# [[Portal:Wiki Scholar]]
# [[Wikiversity:Research]]
&lt;!--
==External links==
# [http://wilderdom.info/wiki/User:James_Neill/Publications James Neill's publications]
--&gt;
[[Category:User:Jtneill/Research| ]]&lt;/noinclude&gt;</text>
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  <page>
    <title>University of Canberra</title>
    <ns>0</ns>
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        <id>10242</id>
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      <comment>Update</comment>
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      <text bytes="6927" sha1="sm4ul5ff1xxxx4swohzoze49ldmhw6c" xml:space="preserve">[[File:Uc-logo.png|right|200px]]
{{TOCright}}
'''{{PAGENAME}}''' (UC) is a mid-size, mid-ranked [[Australian university]] with an emphasis on applied and professional [[learning]] and [[research]]. For more information about this university, see [[w:University of Canberra|University of Canberra]] (Wikipedia). 

[http://www.canberra.edu.au/breakthrough UC's 2013-2017 strategic plan] emphasised developing greater flexible learning and international engagement.

Some UC academics and [[emerging academic]]s have worked or are working towards wider adoption of [[open academia|open academic]] practices such as the use of [[Wikiversity]]. This page lists a range of resources and projects related to this work.

==Purpose==
UC exists for the following purposes[http://www.canberra.edu.au/breakthrough/purpose]:
#To provide education which offers high quality transformative experiences to everyone suitably qualified, whatever their stage of life and irrespective of their origins.
#To engage in research and creative practice which are of high quality and aim to make an early and important difference to the world around us.
#To contribute, through our education and research, to the building of just, prosperous, healthy and sustainable communities which are committed to redressing disadvantage and reconciliation with Australia’s Indigenous peoples.

Everything else - buildings, campuses, structures, titles, hierarchies, pomp, circumstance, (even committees) - may change over time if that is what is required to carry out our purposes.

==People==
These University of Canberra staff accounts have contributed to Wikiversity:
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
&lt;dynamicpagelist&gt;
category=University_of_Canberra/Staff
order=descending
ordermethod=lastedit
mode=unordered
shownamespace=true
&lt;/dynamicpagelist&gt;
&lt;/div&gt;

&lt;gallery&gt;
File:JamesNeillUNH2002OfficeCloseb.jpg|James Neill
File:Ben-rattray.jpg|Ben Rattray
File:Keith.jpg|Keith Lyons
File:Keane-wheeler2.JPG|Keane Wheeler
File:Madepercy.jpg|Michael De Percy
File:Kate-Pumpa.JPG|Kate Pumpa
&lt;/gallery&gt;

==Teaching units on Wikiversity==
Several units taught by [[University of Canberra/Staff|staff at UC]] are available on [[Wikiversity]] in various forms of completeness:
{{center top}}
{| border=1 cellspacing=0 cellpadding=5
|-
! '''[[:Category:University of Canberra/Units|Unit]]''' (current)
! '''[[:Category:University of Canberra/Staff|Convener]]'''
|-
| [[Sport coaching pedagogy]]
| [[User:Postillion|Keith Lyons]]
|-
| [[Exercise and metabolic disease]]
| [[User:Benrattray|Ben Rattray]]
|-
| [[Motivation and emotion]]
| [[User:Jtneill|James Neill]]
|-
| [[Sport event management]]
| [[User:Robinmcconnell|Robin McConnell]]
|-
| [[Sport research]]
| [[User:Benrattray|Ben Rattray]]
|-
| [[Survey research and design in psychology]]
| [[User:Jtneill|James Neill]]
|-
| [[The sport workplace]]
| [[User:Robinmcconnell|Robin McConnell]]
|-
| '''[[:Category:University of Canberra/Units|Unit]]''' (not current or in development)
| '''[[:Category:University of Canberra/Staff|Convener]]'''
|-
| [[Advanced ANOVA]] 
| [[User:Jtneill|James Neill]]
|-
| [[Psychology 102]]
| [[User:Jtneill|James Neill]]
|-
| [[Social Media]]
| [[User:Leighblackall|Leigh Blackall]]
|-
| [[Social psychology (psychology)]]
| [[User:Jtneill|James Neill]]
|-
| [[Government-Business Relations]]
| [[User:Madepercy|Michael de Percy]]
|-
| [[Business, politics and sport]]
| [[User:Leighblackall|Leigh Blackall]]
|-
| [[Composing educational resources]]
| [[User:Leighblackall|Leigh Blackall]]
|-
| [[Using social media for teaching and research]]
| [[User:Leighblackall|Leigh Blackall]] and [[User:Jtneill|James Neill]]
|-
| [[Using the internet for learning and research]]
| [[User:Leighblackall|Leigh Blackall]] and [[User:Benrattray|Ben Rattray]]
|- 
| [[/Wikis for research/]]
| [[User:Leighblackall|Leigh Blackall]]
|}
{{center bottom}}

==X==
Many UC staff, ex-staff, and some students are in this list: https://twitter.com/jtneill/uc-community

==Blogs==
;Current staff
* Lubna Alam - http://lubnalam.wordpress.com/
* Nick Ball - http://archive.is/20121130075134/nick-ball.blogspot.com/
* Stephen Barrass - http://stephenbarrass.wordpress.com
* Michael De Percy - http://www.politicalscience.com.au/
* Sam Hinton - http://meetpi.edublogs.org
* Simon Leonard - http://historyandpresent.blogspot.com/
* Megan Poore - http://meganpoore.com/
* Ben Rattray - http://benrattray.wordpress.com/
* Andrew Read - http://brumbiesteach.blogspot.com/
* Keane Wheeler - http://keanewheeler.blogspot.com/
* Mitchell Whitelaw - http://teemingvoid.blogspot.com/
 
;Past staff
* Leigh Blackall - http://leighblackall.blogspot.com
* Robert Fitzgerald - http://mathetic.info/
* Nicholas Klomp - http://www.canberra.edu.au/blogs/dvce (corporate blog)
* Keith Lyons - http://keithlyons.wordpress.com/
* Danny Munnerley - http://munnerley.com/dan/
* Stephen Parker - http://www.canberra.edu.au/blogs/vc/ (corporate blog)
* Jamie Ranse - http://www.jamieranse.com/

==Recent changes==
{{cot|Recent changes}}
* [[Special:RecentChangesLinked/Category:University_of_Canberra|Recently changes pages  in the University of Canberra category]] of WIkiversity.
* RSS: [http://en.wikiversity.org/w/index.php?title=Special:RecentChangesLinked/Category:University_of_Canberra&amp;feed=atom&amp;target=Category%3AUniversity_of_Canberra%2F  subscribe here].

'''The pages with the most recent changes are listed below:'''
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
&lt;dynamicpagelist&gt;
category=University of Canberra
order=descending
ordermethod=lastedit
mode=unordered
shownamespace=true
&lt;/dynamicpagelist&gt;
&lt;/div&gt;
{{cob}}
==UC Category tree==
{{cot|UC Category tree}}
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;{{#categorytree:{{PAGENAME}}|hideroot|mode=pages}}&lt;/div&gt;
{{cob}}
==Subpages to the UC main page==
{{cot|Subpages}}
{{Special:Prefixindex/{{FULLPAGENAME}}/}}
{{cob}}
==See also==
* [[Wikipedia:University of Canberra|University of Canberra]] (Wikipedia)

==External links==
* [http://www.canberra.edu.au University of Canberra] (Official website)
* [http://www.facebook.com/pages/Canberra-Australia/University-of-Canberra/28782534880? University of Canberra] (Facebook)
* [http://www.youtube.com/results?search_query=+university+of+canberra&amp;aq=f University of Canberra] (Youtube)
* [http://blip.tv/search?q=university+of+canberra University of Canberra] (Blip.tv)
* [http://www.slideshare.net/fsearch/slideshow?q=university+of+canberra&amp;searchfrom=header University of Canberra] (Slideshare)
* [http://www.canberra.edu.au/breakthrough Strategic plan]
&lt;!-- pre-2013 strategic plan
* [http://www.canberra.edu.au/quality/strategic-directions Strategic directions]
* [http://www.canberra.edu.au/learning-teaching/uc-signature-themes Signature themes]
--&gt;

[[Category:University of Canberra]]
[[Category:Real world schools]]</text>
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    <title>Power Generation/Hydro Power/Part2</title>
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&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;color:inherit;&quot;
| {{image|name=Hydroelectric_dam.svg|width=500px|pad=10px|caption=&lt;div style=&quot;font-size:100%;line-height:1.3em;text-align:center;&quot;&gt;'''Figure 2:'''Hydroelectric dam cross-section diagram ( Click on image to view full size image )&lt;/div&gt;|float=left}} 
|rowspan=&quot;2&quot;|
Hydo-electric power stations boast a simple design and construction method that is very robust and reliable(when done properly). The following sub topic discusses the most important constituents of this kind of power station (as shown in figures 2 &amp; 3).


&lt;big&gt;&lt;u&gt;'''Constituents of Hydro-electric power station'''&lt;/u&gt;&lt;/big&gt;
&lt;br&gt;

&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;color:inherit;&quot; 
|- style=&quot;vertical-align:top&quot;
|style=&quot;background-color:#efefef;{{Text color default}};&quot;|&lt;u&gt;'''Reservoir:'''&lt;/u&gt; 
This is where water is stored for use as and when needed. The type and arrangement depends on [[W:Topography|topography]] of the site.
||style=&quot;background-color:#ffdead; {{text color default}};&quot;|&lt;u&gt;'''Penstock:'''&lt;/u&gt; 
This is a conduit (conduits) that carry water to the turbines. They are made of reinforced concrete or steel. A [[W:Surge tank|surge tank]] is installed next to each penstock for over flow control and protection of penstock from bursting.
|-style=&quot;vertical-align:top&quot;
|style=&quot;background-color:#ffdead; {{Text color default}};&quot;|&lt;u&gt;'''Water turbine:'''&lt;/u&gt; 
Water turbines are used to convert hydraulic energy of flowing water into rotational mechanical energy. Figure 3 is an example of the make of a typical water turbine.
|style=&quot;background-color:efefef; {{text color default}};&quot;|&lt;u&gt;'''Generator:'''&lt;/u&gt; 
This machine is used to convert rotational mechanical energy transferred from the turbine through the shaft, into electrical energy. the produced electrical energy is transmitted to the transformer for long distance transmission.
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;


&lt;big&gt;&lt;u&gt;'''Location of Hydro-electric power station:'''&lt;/u&gt;&lt;/big&gt; ''influencing factors''
&lt;br&gt;

&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;color:inherit;&quot; 
|- style=&quot;verticl-align:top&quot;
|style=&quot;background-color:#efefef; {{Text color default}};&quot;|&lt;u&gt;'''Availablility of water:'''&lt;/u&gt; 
Adequate water must be available at good head.
|style=&quot;background-color:#ffdead; {{Text color default}};&quot;|&lt;u&gt;'''Cost and Type of Land:'''&lt;/u&gt; 
Land should be available at reasonable price. The bearing capacity of the land should be enough to withstand huge structures &amp; equipment.
|-style=&quot;vertical-align:top&quot;
|style=&quot;background-color:#ffdead; {{Text color default}};&quot;|&lt;u&gt;'''Storage of water:'''&lt;/u&gt; 
A dam must be constructed to store water in order to deal with variations of water availability during the year.
|style=&quot;background-color:#efefef; {{Text color default}};&quot;|&lt;u&gt;'''Transportaion facilities:'''&lt;/u&gt; 
The site should be accessible by rail and(or) road for ease in transporting equipment &amp; machinery.
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;

This schematic diagram must be properly understood. It is the basis upon which Hydro-electric power station designs are done. The individual power station complexity may differ slightly to the schematic and yet over and above that will use the same principle.

|- 
| {{image|name=Hydroelectric Generator.svg|width=275px|pad=10px|caption=&lt;div style=&quot;font-size:100%;line-height:1.3em;text-align:center;&quot;&gt;'''Figure 3:'''[[W:Water_turbine|Hydraulic turbine]] and [[W:Electrical_generator|Electrical generator]]  diagram ( Click on image to view full size image )&lt;/div&gt;|float=left}} 
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;

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|rowspan=&quot;2&quot;|
Hydo-electric power stations boast a simple design and construction method that is very robust and reliable(when done properly). The following sub topic discusses the most important constituents of this kind of power station (as shown in figures 2 &amp; 3).


&lt;big&gt;&lt;u&gt;'''Constituents of Hydro-electric power station'''&lt;/u&gt;&lt;/big&gt;
&lt;br&gt;

&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;color:inherit;&quot; 
|- style=&quot;vertical-align:top&quot;
|style=&quot;background-color:#efefef; {{text color default}};&quot;|&lt;u&gt;'''Reservoir:'''&lt;/u&gt; 
This is where water is stored for use as and when needed. The type and arrangement depends on [[W:Topography|topography]] of the site.
|style=&quot;background-color:#ffdead; {{text color default}};&quot;&amp;#124;&lt;u&gt;'''Penstock:'''&lt;/u&gt; 
This is a conduit (conduits) that carry water to the turbines. They are made of reinforced concrete or steel. A [[W:Surge tank|surge tank]] is installed next to each penstock for over flow control and protection of penstock from bursting.
|-style=&quot;vertical-align:top&quot;
|style=&quot;background-color:#ffdead; {{Text color default}};&quot;|&lt;u&gt;'''Water turbine:'''&lt;/u&gt; 
Water turbines are used to convert hydraulic energy of flowing water into rotational mechanical energy. Figure 3 is an example of the make of a typical water turbine.
|style=&quot;background-color:efefef; {{text color default}};&quot;|&lt;u&gt;'''Generator:'''&lt;/u&gt; 
This machine is used to convert rotational mechanical energy transferred from the turbine through the shaft, into electrical energy. the produced electrical energy is transmitted to the transformer for long distance transmission.
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;


&lt;big&gt;&lt;u&gt;'''Location of Hydro-electric power station:'''&lt;/u&gt;&lt;/big&gt; ''influencing factors''
&lt;br&gt;

&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;color:inherit;&quot; 
|- style=&quot;verticl-align:top&quot;
|style=&quot;background-color:#efefef; {{Text color default}};&quot;|&lt;u&gt;'''Availablility of water:'''&lt;/u&gt; 
Adequate water must be available at good head.
|style=&quot;background-color:#ffdead; {{Text color default}};&quot;|&lt;u&gt;'''Cost and Type of Land:'''&lt;/u&gt; 
Land should be available at reasonable price. The bearing capacity of the land should be enough to withstand huge structures &amp; equipment.
|-style=&quot;vertical-align:top&quot;
|style=&quot;background-color:#ffdead; {{Text color default}};&quot;|&lt;u&gt;'''Storage of water:'''&lt;/u&gt; 
A dam must be constructed to store water in order to deal with variations of water availability during the year.
|style=&quot;background-color:#efefef; {{Text color default}};&quot;|&lt;u&gt;'''Transportaion facilities:'''&lt;/u&gt; 
The site should be accessible by rail and(or) road for ease in transporting equipment &amp; machinery.
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;

This schematic diagram must be properly understood. It is the basis upon which Hydro-electric power station designs are done. The individual power station complexity may differ slightly to the schematic and yet over and above that will use the same principle.

|- 
| {{image|name=Hydroelectric Generator.svg|width=275px|pad=10px|caption=&lt;div style=&quot;font-size:100%;line-height:1.3em;text-align:center;&quot;&gt;'''Figure 3:'''[[W:Water_turbine|Hydraulic turbine]] and [[W:Electrical_generator|Electrical generator]]  diagram ( Click on image to view full size image )&lt;/div&gt;|float=left}} 
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;

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&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;&quot;
| {{image|name=Hydroelectric_dam.svg|width=500px|pad=10px|caption=&lt;div style=&quot;font-size:100%;line-height:1.3em;text-align:center;&quot;&gt;'''Figure 2:'''Hydroelectric dam cross-section diagram ( Click on image to view full size image )&lt;/div&gt;|float=left}} 
|rowspan=&quot;2&quot;|
Hydo-electric power stations boast a simple design and construction method that is very robust and reliable(when done properly). The following sub topic discusses the most important constituents of this kind of power station (as shown in figures 2 &amp; 3).


&lt;big&gt;&lt;u&gt;'''Constituents of Hydro-electric power station'''&lt;/u&gt;&lt;/big&gt;
&lt;br&gt;

&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;&quot; 
|-valign=&quot;top&quot;
|bgcolor=#efefef|&lt;u&gt;'''Reservoir:'''&lt;/u&gt; 
This is where water is stored for use as and when needed. The type and arrangement depends on [[W:Topography|topography]] of the site.
|bgcolor=#ffdead|&lt;u&gt;'''Penstock:'''&lt;/u&gt; 
This is a conduit (conduits) that carry water to the turbines. They are made of reinforced concrete or steel. A [[W:Surge tank|surge tank]] is installed next to each penstock for over flow control and protection of penstock from bursting.
|-valign=&quot;top&quot;
|bgcolor=#ffdead|&lt;u&gt;'''Water turbine:'''&lt;/u&gt; 
Water turbines are used to convert hydraulic energy of flowing water into rotational mechanical energy. Figure 3 is an example of the make of a typical water turbine.
|bgcolor=#efefef|&lt;u&gt;'''Generator:'''&lt;/u&gt; 
This machine is used to convert rotational mechanical energy transferred from the turbine through the shaft, into electrical energy. the produced electrical energy is transmitted to the transformer for long distance transmission.
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;


&lt;big&gt;&lt;u&gt;'''Location of Hydro-electric power station:'''&lt;/u&gt;&lt;/big&gt; ''influencing factors''
&lt;br&gt;

&lt;div style=&quot;float:center&quot;&gt;
{| border=&quot;0&quot; style=&quot;background:transparent;&quot; 
|-valign=&quot;top&quot;
|bgcolor=#efefef|&lt;u&gt;'''Availablility of water:'''&lt;/u&gt; 
Adequate water must be available at good head.
|bgcolor=#ffdead|&lt;u&gt;'''Cost and Type of Land:'''&lt;/u&gt; 
Land should be available at reasonable price. The bearing capacity of the land should be enough to withstand huge structures &amp; equipment.
|-valign=&quot;top&quot;
|bgcolor=#ffdead|&lt;u&gt;'''Storage of water:'''&lt;/u&gt; 
A dam must be constructed to store water in order to deal with variations of water availability during the year.
|bgcolor=#efefef|&lt;u&gt;'''Transportaion facilities:'''&lt;/u&gt; 
The site should be accessible by rail and(or) road for ease in transporting equipment &amp; machinery.
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;

This schematic diagram must be properly understood. It is the basis upon which Hydro-electric power station designs are done. The individual power station complexity may differ slightly to the schematic and yet over and above that will use the same principle.

|- 
| {{image|name=Hydroelectric Generator.svg|width=275px|pad=10px|caption=&lt;div style=&quot;font-size:100%;line-height:1.3em;text-align:center;&quot;&gt;'''Figure 3:'''[[W:Water_turbine|Hydraulic turbine]] and [[W:Electrical_generator|Electrical generator]]  diagram ( Click on image to view full size image )&lt;/div&gt;|float=left}} 
|}&lt;/div&gt;

&lt;br style=&quot;clear: both;&quot;&gt;

{{CourseCat|filing=deep}}</text>
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  <page>
    <title>Comparative law and justice/Jordan</title>
    <ns>0</ns>
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    <revision>
      <id>2804320</id>
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      <timestamp>2026-04-11T12:59:10Z</timestamp>
      <contributor>
        <username>CarlessParking</username>
        <id>3064444</id>
      </contributor>
      <comment>/* Economic Development (2008) */</comment>
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      <text bytes="21059" sha1="867eozrfvsloozgss56f8t6eo3blksu" xml:space="preserve">Part of the [[Comparative law and justice]] Wikiversity Project
{{Comparative law and justice project|region=Asia}}
== Basic Information ==
[[image:Flag_of_Jordan.svg|225px|right|Jordanian Flag]]             

'''Estimated population(2009):''' 
{| border=1 cellspacing=5 cellpadding=5
|Total population
|Male and Female
|6,342,948
| 100.0%
|- 
|Male 00-14yrs
|1,014,183 
|Female 00-14yrs
|973,538
|31.3%
|-
|Male 15-64
|2,183,638
|Female 15-64
|1,904,420
|69.5%
|-
|Male 65+
|128,759
|Female 65+
|138,410
|4.0%
|}
&lt;ref&gt;Central Intelligence Agency. 2009. The World Factbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;


'''The People in Jordan'''

People in Jordan have a Jordanian Nationality. Jordan is predominanitely arab(98%), there is a very small group of Circassian and Armenian which is about 2% total.&lt;ref&gt;Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;
 
In the main city lives a mix of people. City life is completely different than the rural areas. Amman is split into Eastern Amman and Western Amman. Eastern Amman is the more conservative part of Amman. Western Amman is the more liberal side of the country. &lt;ref&gt;Personal observations&lt;/ref&gt;


'''Religion'''

The major religion is Islam, Jordan is about 92% Sunni Muslims, there are other religious groups.Christian make up 6% of Jordan, the majority are Greek Orthodox, but some are Roman Catholics, Syrian Orthodox, Armenian Orthodox and Protestant. There is a very small group of Shia Muslims and Druze which are about 2% of the population.
&lt;ref&gt;Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;


'''Language'''

Arabic is the first language used in Jordan. English is widely used and is the official second language. English is mostly spoken by upper and middle class people. French is also taught in some schools as a third language. 
&lt;ref&gt;Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;


===Economic Development (2008) ===
Jordan has a GDP (purchase Power) of $31.61 Billion
GDP per capita (PPP) $5,100. Jordan is popular for exporting clothing, fertilizers, potash, phosphate mining and pharmaceuticals.Jordan also works on petroleum refining, cement, inorganic chemicals, light manufacturing,and is a great tourist attraction.Jordan relies on Saudi Arabia for the majority of their imports (20.3%). Jordan also imports machinery, transportation equipment, iron and cereals from China, Germany and the U.S. (2008). Jordan's currency is the Jordanian Dinar, the exchange rate of the Jordanian Dinar (JOD) is approximately 0.709 U.S dollar. 
&lt;ref&gt;Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009. https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;


[[Image:Petra Jordan BW 21.JPG|thumb|180px|right|Petra is a great tourist attraction in Jordan.]]

===Health, and Education===
The total infant mortality is 14.97 deaths/1,000 live births of this number 17.91 are males and 11.86 are females(2009). The life Expectancy in Jordan is 78.87 years old, males live to approximately 76 years old and females live up to 81 years old. Fertility rate in Jordan is 2.39 children born/woman (2009 est)
ref&gt;Central Intelligence Agency. 2009. &lt;ref&gt; The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;

===Brief History and Geography===
                                                                  
   [[Image:amman_location.png|thumb|180px|left| Map of Jordan]]

The Hashemite Kingdom of Jordan is located in the Middle East. It is located in the northwest of Saudi Arabia. 
&lt;ref&gt;Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;
Jordan is bordered with Palestine, Syria, Lebanon, Iraq and Saudi Arabia.
Jordan became a British protectorate in 1921, that was established as the Emirate of Transjordan following the dissolution of the Ottoman Empire in WWI. Jordan gained it's independence in 1946 and became the Hashemite Kingdom of Jordan in 1950. Jordan was governed by King Hussein (1953-1999). In 1967, Jordan lost the West Bank and Jordan renounced its claims to the West Bank in 1988. In 1989, King Hussein created parliamentary elections. Political parties were created in 1992. After a couple of years in 1994, King Hussein signed a peace treaty with Israel with the help of the United States to settle the Arab Israeli struggle. King Hussein died in 1999 and his son King Abdullah II, took over the monarchy. After the war on Iraq, Jordan took in thousands of Iraqis who had no place to go but Jordan.The parliament now has more women than ever (20% of parliament are reserved to women). King Abdallah is now focusing on the socioeconomics of the country, improving education and developing a better healthcare system.

&lt;ref&gt; Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;

==Jordan's Legal System==
Jordan has a mixed legal system based on civil law, Sharia Law (Islamic Law)and Customary law.
&lt;ref&gt;The University of Ottowa. &quot;Muslim Law Systems and mixed systems with a Muslim Law Tradition. Website accessed 12/02/2009.http://www.juriglobe.ca/eng/sys-juri/class-poli/droit-musulman.php&lt;/ref&gt;
On April 1928, the Constitution was created.In 1929, Jordan had its first election. In 1946, Jordan got her independence from Brittain, the king then adopted a new constitution in 1947. In 1952 the constitution was ratified by King Talal.&lt;ref&gt;The University of Ottowa. &quot;Muslim Law Systems and mixed systems with a Muslim Law Tradition&quot;.Website accessed 12/02/2009.http://www.juriglobe.ca/eng/sys-juri/class-poli/droit-musulman.php&lt;/ref&gt;

'''''Jordanian Constitution'''''&lt;ref&gt;&quot;The Constitution of The Hashemite Kingdom of Jordan.&quot; A Living Tribute to the Legacy of King Hussein I. Web. 05 Dec. 2009. &lt;http://www.kinghussein.gov.jo/constitution_jo.html&gt;.&lt;/ref&gt;

Administrative Divisions in Jordan consists of 12 governorates (muhafazat, singular - muhafazah); Ajlun, Al 'Aqabah, Al Balqa', Al Karak, Al Mafraq, 'Amman, At Tafilah, Az Zarqa', Irbid, Jarash, Ma'an, Madaba.
&lt;ref&gt;Central Intelligence Agency. 2009. The World Fatbook, &quot;Country Comparison: Population.&quot; Website accessed 09/24/2009,
https://www.cia.gov/library/publications/the-world-factbook/geos/jo.html&lt;/ref&gt;

===Executive Branch===
Head of the State: King Abdallah II
The king must be sane, male Muslim, the son of Muslim parents, and born of a lawful wife.The king has the most powerful position in the government he:appoints the prime minister, the president and members of the senate, judges and other senior government and military functionaries. the king is also the commander and cheif of the army, the king approves and promulgates the laws, declares war, concludes peace, and signs treaties (which in theory must be approved by the National Assembly).The king also convenes, opens, adjourns, suspends, or dissolves the legislature; he also orders, and may postpone, the holding of elections.The king has veto power that can be overridden only by a two-thirds vote of each house.The head of government is the Prime Minister Nader al-DAHABI
and the Cabinet is appointed by the prime minister in consultation with the king.
&lt;ref&gt;Metz, Helen, ed. Jordan: A Country Study. Washington: GPO for the Library of Congress 1989. Jordan-The Government.Website accessed 10/16/2009.http://countrystudies.us/jordan/54.htm&lt;/ref&gt;

===Elections===
The Monarch is not elected, he takes over the kingdom as a Royal succession by a male descent in the Hashimite Family. The oldest son gets to hold the throne after the death of his father. If the king dies without ever having a son his oldest brother has seniority. followed by the eldest son of the other brother according to their, again according to the oldest. &quot;If their is no suitable direct heir, then the National Assembly select a successor from the descendants of the founder, King Hussein Bin Ali.&quot; &lt;ref&gt;&quot;Jordan - THE GOVERNMENT.&quot; Country Studies. Web. 05 Dec. 2009. &lt;http://countrystudies.us/jordan/54.htm&gt;.&lt;/ref&gt;
There are no elections held for the Prime Minister, he is appointed by the monarch.
&lt;ref&gt;&quot;Jordan - THE GOVERNMENT.&quot; Country Studies. Web. 05 Dec. 2009. &lt;http://countrystudies.us/jordan/54.htm&gt;.&lt;/ref&gt;
In the late 1970's the Executive branch suspended parliament. The Executive took over the powers of the legislative branch. This lasted until 1984, this was called the ninth house of representatives. They ruled up until 1989. There were no party affiliations at this time.
&lt;ref&gt;Jordan - THE GOVERNMENT.&quot; Country Studies. Web. 05 Dec. 2009. &lt;http://countrystudies.us/jordan/54.htm&lt;/ref&gt;

===Legislative Branch===
Jordan has a Bicameral Legislature called the National Assembly (Majlis Al-Umma), it is composed of the House of Representatives (Majlis Al-Nuwwab)and the Senate (Majlis Al-Ayan). The king appoints the senate members, consisting of fifty-five members for a four year term. The House of Representatives had 80 seats prior to 2001, now it consists of 110 members, 6 seats reserved for women, 9 for Christians, 3 for Circassions and 9 for Jordanian Bedouins. All candidates have to be 30 years of age or older. Elections are not mandatory and any Jordanian citizen 18 years or older can vote. Voters out come in the 2003 election was 58%, which is very interesting.

&lt;ref&gt;MENA Election Guide - Amman Jordan. Web. 05 Dec. 2009. &lt;http://www.mena-electionguide.org/&gt;.&lt;/ref&gt;

=='''Courts'''==
Jordan has three court systems, Civil courts, Military courts, and religious courts. Civil courts contain the magistrate's courts,courts of first instance, major felonies courts, courts of appeal and the court of Cassation (Supreme Court). The Magistrate's courts deal with criminal cases. They also deal with civil suits that are not exceeding JD750, they usually have one judge. In Amman, the Capital there are 14 magistrates in other cities 2-3 magistrates each. The courts of first instance hear cases that fall outside the Magistrate's courts jurisdiction. They also hear all criminal appeal cases that have a prison sentence of 2-3 weeks as well as any civil suits that are exceeding JD750. The major felonies court hears cases that have a prison sentence over than three years. This court hears cases that deal with murder, manslaughter, rape and sexual assaults, there is only one court and is located in the major city Amman. Three judges preside appeals are made to the Court of Cassation. Courts of Appeal hear all the cases brought by the magistrate's courts and the courts of first instance, three judges preside over these courts. the court of Cassation has fifteen judges but normally five judges hear the cases. They deal with jurisdiction issues, it also hears cases that deal with habeas corpus petitions. 
Military Courts deal with any crimes that affect the national security of the country. The cases that are heard include drugs or weapon smuggling, Espionage and military personnel. 
Religious courts have jurisdiction over all family matters, marriage, divorce, adoption and custody. Jordan does not allow civil marriages, therefore they are not performed in civil courts. There are religious courts for Muslims based on Sharia'a Law (Islamic Law) and there are courts based on Christianity. 
There are several other courts Juvenile courts, police court, land settlement courts, income tax court, customs court and tribal courts.
Jordan does not have a system that uses Judicial review. Since Jordan is a civil Law country, it has an Inquisitorial system. Inquisitorial system is different from our system in the U.S. Jury is not present in Inquisitorial systems. The judges are trained civil servants.The trial is to seek the truth, no competition. In this type of system, lawyers play a very passive role.The judge plays a very important role in this system.As a Muslim country, Judges in Jordan have studied Sharia in depth before becoming judges. 
&lt;ref&gt;&quot;Jordanian Legal System - U.S. Embassy Amman, Jordan.&quot; Embassy of the United States Amman, Jordan - Home. Web. 05 Dec. 2009. &lt;http://jordan.usembassy.gov/acs_jordanian_legal_system.html&gt;.&lt;/ref&gt;

===Punishment===
Punishment in Jordan is used as a deterrence, to keep others from committing similar crimes. Severity of punishment will keep people from committing crimes. Jordan is one of many countries that use the death penalty for crimes such as murder or Espionage. Life prison sentences were imposed for felonies that are intended on negatively affecting the national security, homicide that results from beating or hitting someone, and serious forms of theft. Shorter imprisonment was prescribed for these same offenses if mitigating circumstances warranted. Terrorist activity and membership in terrorist organizations, counterfeiting, forgery of official documents, and abduction are also punishable by prison.''very limited information'' &lt;ref&gt;&quot;Jordan Criminal Code - Flags, Maps, Economy, History, Climate, Natural Resources, Current Issues, International Agreements, Population, Social Statistics, Political System.&quot; Photius Coutsoukis; Photius; Photios; Fotis Koutsoukis. Web. 05 Dec. 2009. &lt;http://www.photius.com/countries/jordan/national_security/jordan_national_security_criminal_code.html&gt;&lt;/ref&gt; 
Jordan also holds some show trials especially when it is a crime of treason or espionage. A person who committed Espionage or treason will be hung in the middle of the down-town.&lt;ref&gt;personal observations&lt;/ref&gt;

Misdemeanors include,gambling in public places, bribery, perjury, simple forgery, slander, embezzlement, assault and battery, and disturbing the peace. There are some crimes that are made criminal because they violate Sharia, desertion of a child, abortion, marrying a girl under the age of sixteen, openly ridiculing the Prophet Muhammad (PBU), and breaking the fast of Ramadan. Most non-criminal acts are fined. A person who has committed a crime will have a record for six years then he/she is allowed to have his record expunged. 
&lt;ref&gt;&quot;Jordan Criminal Code - Flags, Maps, Economy, History, Climate, Natural Resources, Current Issues, International Agreements, Population, Social Statistics, Political System.&quot; Photius Coutsoukis; Photius; Photios; Fotis Koutsoukis. Web. 05 Dec. 2009. &lt;http://www.photius.com/countries/jordan/national_security/jordan_national_security_criminal_code.html&gt;&lt;/ref&gt;

Jordan has three prisons, two of which have been shut down, the main prison in Juwaidah holds about 5,448 prisoners as of 2002. (''very limited information'')
&lt;ref&gt; &quot;Prisons and prison systems: a global ... -.&quot; Google Books. Web. 05 Dec. 2009. &lt;http://books.google.com/books?id=RTH31DgbTzgC&amp;pg=PA149&amp;lpg=PA149&amp;dq=prison+rates+in+amman&amp;source=bl&amp;ots=1bsdDbUobj&amp;sig=u3Mop5MzPGMdXV9KeCMF&gt;. &lt;/ref&gt;

===Law Enforcement, Crime Rates and Public Opinion===
Police in Jordan is run by the army. The head of the police is the army general. Jordanian police is Proactive they are all over the streets looking for trouble. People are trained in AL-Zarqa Police Academy, no specific time is provided, but in neighboring countries it is 3 months. Jordan has a Centralized police structure, one national police force, enforces the laws. (''i had a very hard time finding information on police in Jordan''
&lt;ref&gt; &quot;Prisons and prison systems: a global ... -.&quot; Google Books. Web. 05 Dec. 2009. &lt;http://books.google.com/books?id=RTH31DgbTzgC&amp;pg=PA149&amp;lpg=PA149&amp;dq=prison+rates+in+amman&amp;source=bl&amp;ots=1bsdDbUobj&amp;sig=u3Mop5MzPGMdXV9KeCMF&gt;. &lt;/ref&gt;
Crime rates are very low in Jordan in comparison to other countries. In the table provided, there is a comparison between Jordan, U.S.A and Sweden. As seen in the table, Jordan has a substantially lower crime rate than both the U.S and Sweden. Some crimes maybe under reported, but this is a substantial difference.As a person that has lived in Jordan for many years assert that Jordan has a very low crime rate. The reason behind this is predominantly the people's religious views, also the culture which emphasizes social bonds; therefore the low crime rates.  

'''&lt;big&gt;Number of Crimes Per 100,000 Citizens&lt;/big&gt;'''
{| border=1 cellspacing=5 cellpadding=5
| '''Category'''
| '''U.S.A'''
|'''Sweden'''
|'''Jordan'''
|-
|'''Total Offenses'''
|537.4
|12,620.3
|897.5
|-
|'''Murder'''
|8.9
|9.5
|6.9
|-
|'''Rape'''
|39.2
|20.6
|0.9
|-
|'''Theft (all kinds)'''
|489.1
|7410
|160.4
|-
|'''Violent Theft'''
|237.71
|60.5
|1.1
|-
|'''Auto Theft'''
|591.2
|616.1
|15.7
|-
|'''Drug Offenses'''
|No Data
|358.48
|6.7
|-
|&lt;small&gt;Source: International Crime Statistics. International Criminal Police Association (INTERPOL), Lyon, France, 1994.&lt;/small&gt;
|}

&lt;ref&gt;&quot;Jordan - Government - The Judicial Branch.&quot; A Living Tribute to the Legacy of King Hussein I. Web. 05 Dec. 2009. &lt;http://www.kinghussein.gov.jo/government4.html&gt;.&lt;/ref&gt;

The death penalty is highly supported in Jordan in Murder cases, treason and Espionage. Some people, especially people in the rural areas support honor killings. People in Jordan do not have a lot of faith in the police, there is a popular saying &quot;the countries defenders are the countries thieves&quot;.

===Family Law===
Based on Sharia Law (Islamic Law)the right age for marriage is when the male and female hit puberty. since everyone reaches puberty at different age Jordan adopted a set age for marriage. The Legal marital age is fifteen for females, and sixteen for males, but must have parental consent. Anyone under the age of 18 must get parental permission for marriage. Polygamy is allowed in Jordan, but there are some restrictions. The man must provide separate houses for each wife. He must treat all the wives equally. He must have no more than four wives and must declare his marital status. Judicial divorce is allowed from either side, man or woman. A woman gets child custody until puberty.&lt;ref&gt;Jordan, Hashemite Kingdom of.&quot; Emory Law: More Than Practice: Home. Web. 05 Dec. 2009. &lt;http://www.law.emory.edu/ifl/legal/jordan.htm&gt;.&lt;/ref&gt; 
Inheritance laws are based on Sharia'a Law, Males get twice as much as the females. (males are the care takers therefore they get more). The male is the head of the household. Women have rights in Jordan just like any other man legally, but things are different in practice. Women have the right to vote. Jordan is the first country in the middle east to allow women in the police force and in the army.
&lt;ref&gt;Jordan, Hashemite Kingdom of.&quot; Emory Law: More Than Practice: Home. Web. 05 Dec. 2009. &lt;http://www.law.emory.edu/ifl/legal/jordan.htm&gt;.&lt;/ref&gt;

===Human Rights and Social Inequality===
Jordan is part of many human rights programs including the International Covenant on Civil and Political Rights, the International Covenant on Economic, Social and Cultural Rights and the Convention against Torture and Other Cruel, Inhuman or Degrading Treatment or Punishment.
&lt;ref&gt;&quot;| Middle East - Jordan.&quot; Human Rights First. Web. 05 Dec. 2009. &lt;http://web.archive.org/web/20070629145037/http://www.humanrightsfirst.org/middle_east/jordan/hrd_jordan.htm&gt;.&lt;/ref&gt; but Jordan does not follow any of them.  Freedom of Speech, association and Expression is also limited in Jordan, especially political speech. People can suffer a lot if they make any kind of critic towards the monarchy or public political figures. 

In practice Jordan has specific discrimination, Jordanians are treated differently than non-Jordanians. Jordanians get specific treatments, for example, Jordanian high school graduates go to public universities even if they score really bad on their final exams whereas others have to do real well to be able to go to public colleges. This may not be in the Laws but it is done in practice.&lt;ref&gt;self observations&lt;/ref&gt;
There are obvious class struggles in Jordan, poor, middle and upper class. There is a huge gap, the middle class is really small. the majority of the country is the lower class who live in really bad poverty. This is very clear from the way of life in Jordan.&lt;ref&gt;self observations&lt;/ref&gt; 
when it comes to discrimination, there is a great deal of discrimination from the Jordanian descendants towards palestinian descendants, even though they all carry the Jordanian citizenship they are not all treated equally.
&lt;ref&gt;self observations&lt;/ref&gt;

==Works Cited==
&lt;references /&gt;

[[Category:Jordan]]</text>
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    <title>Comparative law and justice/Canada</title>
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      <id>2804456</id>
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      <timestamp>2026-04-12T10:03:00Z</timestamp>
      <contributor>
        <username>14gulaman</username>
        <id>3065580</id>
      </contributor>
      <comment>cleaned up a few things, clarified the lawyer part, tried my best not to get an aneurysm</comment>
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      <text bytes="26923" sha1="m56y78ar5654qeum9a19yc9qu2bdmwn" xml:space="preserve">Part of the [[Comparative law and justice]] Wikiversity Project
{{Comparative law and justice project|region=North America}}
{{Wikipedia|Canada}}
[[Image:Flag_of_Canada.svg|100px|Canadian Flag]]
== Basic Information ==
Canada's population is 33,487,208, of that population it consists of many different races. British Isles origin consists of 28% of the population, French origin make up 23%, other European make up 15%, Indian and Inuit make up 2%, 6% is made up for Asian, African and Arab, and then 26% is mixed back grounds. The official language consist of two languages, English 59.3%, French 23.2%, and the other 17.5 % is a mix of other languages. In this mix of diversity there is also a mix of religions.43% of the population is Roman Catholic, 23% is Protestant, 4% is Christian, 2% is Muslim, and the other 16% is none.&lt;ref name=&quot;infoplease&quot;&gt;http://www.infoplease.com/ipa/A0107386.html?pageno=2&lt;/ref&gt;

[[Image:Canadamap.png|thumb|right|alt=Map of Canada|Map of Canada]]
Aaron Latham 
Canada is located The Geography of Canada is that it covers most of the North American Continent, it is larger than the United States. Canada's geography varies depending on what direction you go in. To the east are the Maritime Provinces. These provinces have an irregular coastline on the Gulf of the St. Lawrence and the Atlantic. The St. Lawrence River alone covers most of southern Quebec and Ontario. To the West heading towards the Pacific, British Columbia, Yukon and Alberta are covered with many mountain ranges. The Pacific border is filled with different channels and fjords.&lt;ref name=&quot;infoplease&quot;&gt;http://www.infoplease.com/ipa/A0107386.html?pageno=2&lt;/ref&gt;
Canada's locations is Northern North America, bordering the North Atlantic Ocean to the east the North Pacific Ocean to the west, to the north you will find the Artic Ocean. the Geographic Coordinates are 60 00N 95 00 W. The area in total is 9,984,670 square kilometers. The climate of Canada ranges but mostly it is temperate in the South and subarctic in the north. The Arable Land available in Canada is 4.57%. Canada ranks second largest in the world, the first being Russia. The Vast majority of the population is concentrated within 160 km of the United States border. &lt;ref name=&quot;cia info&quot;&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/ca.html&lt;/ref&gt;

===Economic Development, Health, and Education===

Canada's median age is 40.4 years old and the birth rate is 10.3 for every 1,000, the death rate is 7.74 to every 1,000. Life expectancy in Canada is age 81.23, the breakdown is 78.69 for males and 83.91 for females. People living with HIV in Canada is 73,000 which is ranked 55th in the world. The school expectancy is 17 years of education.&lt;ref name=&quot;cia info&quot;&gt;https://www.cia.gov/library/publications/the-world-factbook/geos/ca.html&lt;/ref&gt;

The infant mortality rate is 5.0/1000. The life expectancy rate is 81.2 . The literacy rate in Canada is 99%. The birth rate is 10.2/1000 . The total population of Canada is 33,387,208 people.&lt;ref name=&quot;info please&quot;&gt;http://www.infoplease.com/ipa/A0107386.html?pageno=2&lt;/ref&gt;

Canada has a GDP of $1.266 trillion a year, the GDP per Capita is $38,400. Its exports include: motor vehicles and parts, industrial machinery, aircraft, telecommunications equipment, chemicals, plastics, fertilizers, wood pulp, timber, crude petroleum, natural gas, electricity and aluminum. Canada's imports: machinery and equipment, motor vehicles and parts, crude oil, chemicals, and durable consumer goods. The key industries in Canada are: transportation, equipment, chemicals, processed/unprocessed minerals, food products, paper products, fish products, petroleum and the natural gas.

===Brief History===
Canada obtained independence over years of constitutional changes. Canada was founded by the Vikings by accident, they were making a sailing mission from Iceland to Greenland when they were blown off track to find what we know as today Canada. John Cabot was the person who gave England the rights to Canada in his findings, he found the waters were rich with fish. Canada was discovered by France because of Jacques Cartier.&lt;ref name=&quot;info please&quot;/&gt; Jacques Cartier left his followers to return back to France turning his back on the Canadian Project. For almost 60 years France ignored the Canadian project until Champlain came along. Champlain pleaded with his leader to allow him to go and settle on the St. Lawrence (Quebec). Quebec was settled in 1604.&lt;ref name=&quot;canadian history&quot;&gt;http://www.cyber-north.com/canada/history.html&lt;/ref&gt;

One of the largest cities in Canada is Montreal. The city itself was started for a mission. This city was the basis for the Canadian Missionary movement. Catholics set up posts all around this area in an attempt to convert the Hurons. The Hurons being a Native American tribe were being threatened by the Iroquois. The Christians attempted to help the Hurons but both were killed in the Iroquois final invasion in Huronia. The Iroquois tribe posed as a threat to all settlers until finally in 1660 Adam Dollard des Ormeaux led a group of men to defend the land. Only few Frenchmen died but many Native Americans did, the Iroquois retreated and were no longer a threat. &lt;ref name=&quot;canadian history&quot;&gt;http://www.cyber-north.com/canada/history.html&lt;/ref&gt;

Jean Talon played a very important part in Canadian History. In the beginning Canada did not resemble anything of what could be considered a democratic government. The job of intendant was to run things in the territory. Jean Talon became intendant in 1665, he encouraged great changes in &quot;New France&quot;. He encouraged business, agriculture, crafts, exploration of the land and immigration. In 1666 Talon took a census of the land, he found that the population of &quot;New France&quot; was only 3,215. England had gained control of more land and had a larger population. The rivalry was then started between France and England.&lt;ref name=&quot;canadian history&quot;&gt;http://www.cyber-north.com/canada/history.html&lt;/ref&gt;

England saw the money that the French were making because of the fur trade and decided that they also wanted to involved in this endeavor. They took over the Hudson Bay and it's resources to help better their trade. The French for years threatened them until 1686 when they finally invaded and regained control of the Hudson Bay territory. England and France signed a peace treaty in 1713, but the rivalry was still there and then another fresh war broke out between the two for control of &quot;New France&quot;. In the midst of things a strong British Force stepped in and gained control. This started the early British Rule.&lt;ref name=&quot;canadian history&quot;&gt;http://www.cyber-north.com/canada/history.html&lt;/ref&gt;

==Governance==
Canada has a government that would be labeled as a constitutional monarchy, and it is governed by its own house of commons.  Canada has a constitution that consists of written acts, customs and traditions. A part of it is the Constitution Act of 29 March 1867. This specific act made a federation with four provinces. Another important act of the Canadian Constitution  is Act 17 April 1982. This Act transferred over power from Britain to Canada, along with this shift of power they added a &quot;Canadian Charter of Rights and Freedom&quot;.&lt;ref name=&quot;travelblog&quot;&gt;http://www.travelblog.org/world.ca-gov.html&lt;/ref&gt;

The government of Canada is organized with 3 branches. The Executive branch which is headed by a monarch, this position is hereditary, there is no elections for this position. Also in this branch is the Governor General, who is appointed by the monarch. Then there is the Prime Minister who is mainly there for advice on issues, who is the leader of the major party of the winner in the legislative elections. The second branch of the Canadian Government is the Judicial Branch. This branch is made up of the Supreme Court Of Canada, the judges are appointed by the Governor General with advice from the Prime Minister. The third branch is the Legislative Branch. This Branch is a bicameral Parliament. The two parts of the Parliament are the Senate and the House Of Commons. The Senate is whom ever the Governor General appoints, they serve till they are 75 years of age. The House of Commons are members elected by direct, popular vote and they are to serve up to five years. It is common in Canada for the Liberal Party to hold the majority of the votes in the house of commons. Government officials mostly get their positions through the Governor General with the advice of the Prime Minister.&lt;ref name=&quot;travelblog&quot;/&gt;

===Elections===
Canada has elections at least every five years. The elections are always by secret ballot. In order to vote in a Canadian election you must be 18 years of age and a Canadian citizen.&lt;ref name=&quot;goingtocanada&quot;&gt;http://www.goingtocanada.gc.ca/CIC/display-afficher.do?id=0000000000013&amp;lang=eng&lt;/ref&gt; In Canada by law there has to be a general election held on the third Monday of the fourth calendar year following a general election. In charge of handling all of matters related to elections is the Chief Electoral Officer. The Chief is appointed by the house of commons, so that every political group is represented. The Chief only answers to Parliament and will only take orders from Parliament, this sets and equal setting for all political parties involved in the elections.  &lt;ref name=&quot;elections&quot;&gt;http://www.elections.ca/content.asp?section=loi&amp;document=index&amp;dir=2007&amp;lang=e&lt;/ref&gt;

A recent law passed that states you not only have to be 18 years old but also have to prove your identity. When a voter is going to vote they must have a government issued form of identification, and they also have to have proof of address. If a voter cannot provide this information on the spot then they may take an oath and have someone who has their identification and proper paper work with them to vouch for them. It is public knowledge of who chooses to exercise their right to vote.&lt;ref name=&quot;elections&quot;&gt;http://www.elections.ca/content.asp?section=loi&amp;document=index&amp;dir=2007&amp;lang=e&lt;/ref&gt;

===Judicial Review===

In Canada there is plenty of room for Judicial Review. The Canadian Supreme Court hears special cases and has two crucial decisions with every case. The Supreme court can grant review to a can and one on the merits. &lt;ref name=&quot;court&quot;&gt;http://www.bsos.umd.edu/gvpt/lpbr/subpages/reviews/flemming704.htm&lt;/ref&gt;  Judicial Review isn't as easy as is in other countries, in Canada is more complex and controversial. Canada adopted the charter of rights and freedoms which in it self is Canada's judicial review. Before the adoption of the Charter the courts reviewed legislation and decided whether it was to go on to federal court or not, the charter now takes care of the guidelines for Canada taking out one step of the process, the case either goes to court or it doesn't based on what the Charter states. &lt;ref name=&quot;review&quot;&gt;http://www.uregina.ca/sipp/conference_2007/english/abstracts/M_Harrington.pdf&lt;/ref&gt;

==Courts and Criminal Law==
The levels of courts in Canada start at the Provincial Court. This is were the majority of cases are dealt with, there are many levels but the two main courts are the Provincial Court then the British Columbia Supreme Court. Only for serious crimes and cases does the Supreme Court hear the case. The majority of cases are dealt with in the Provincial Court where there is a Judge that has been appointed by the Province, and the Judge sits alone without a jury. If the case is serious enough then the accused most often has the right to a trail before a Judge alone. The jury are made up of around 12 people specifically chosen by the lawyer of the accused. The lawyer hand picks non-bias and truthful citizens. The jury are given facts about the law, then they are instructed to use that information and apply to what they hear and court, and determine what the outcome is.&lt;ref name=&quot;canadalegal&quot;&gt;http://www.canadalegal.info/ref-canada-criminal-law/canada-criminal-law-courts.html&lt;/ref&gt;

The courts in Canada are organized in levels. First being Provincial or Territorial courts, the nest being the Courts of Appeals, the next is Federal Court, after that Canada has two specific federal courts which are the Tax Court of Canada then The Military Court, after all of these there is the Supreme Court of Canada.&lt;ref name=&quot;Canada's court&quot;&gt;http://www.justice.gc.ca/eng/dept-min/pub/ccs-ajc/&lt;/ref&gt;

The key factors in the Court system of Canada are the Court itself and the Judge. Many times there isn't a jury to hear the case if it is not of a level of seriousness. Usually in Canada if you commit a crime at the Provincial Level then you are to appear before a judge for sentencing. If the case does has high value and goes to the Federal courts then a lawyer is a factor in the court system. There is also a jury if the case is heard at the Federal level, they are different from the Jury of the United States because this jury is handed all the facts, they are taught what the law is and how the case is applied to it. In this sense the court sets the person accused for failure.

In order to practice law in Canada, it is the same process as other nations such as the United States. You must pass the [[wikipedia:Law_School_Admission_Test|Law School Admission Test]] and apply to and complete a provincially/nationally-approved law program in order to get their law degree. Most aspirants complete another program at a post-secondary school before applying to a Law School, typically to gain valuable skills and experience needed to complete the Law School Admission Test, or in order to fulfill prerequisite requirements for their program of choice, as the Bachelor of Laws is practically non-existent.&lt;ref&gt;{{Cite web|url=https://www.lawsociety.ab.ca/lawyers-and-students/become-a-lawyer/application-admission/articling-process/|title=Articling Process|website=Law Society of Alberta|language=en-US|access-date=2026-04-12}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://nca.legal/|title=National Committee on Accreditation (NCA)|website=Federation of Law Societies of Canada|language=en-US|access-date=2026-04-12}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://law.ucalgary.ca/future-students/high-school-students|title=Info for High School Students {{!}} Faculty of Law {{!}} University of Calgary|website=law.ucalgary.ca|language=en|access-date=2026-04-12}}&lt;/ref&gt;

After achieving a law degree, you will have to be called to the bar in the province in which you seek to practice in. Most Law Societies require Law Students to complete [[wikipedia:Articled_clerk|Articling]] and to complete professional development training and education in order to prepare them to practice law before being admitted to the Law Society.
===Punishment===
Many offenses in Canada that are labeled as &quot;Summary Offence&quot; end in the sentence of 6 months maximum imprisonment and $2,000 in fines from the government.&lt;ref name=&quot;canadalegal&quot;&gt;http://www.canadalegal.info/ref-canada-criminal-law/canada-criminal-law-courts.html&lt;/ref&gt; In Canada the punishment for rape is life imprisonment and a chance of parole after 25 years.&lt;ref name=&quot;punishment rape&quot;&gt;http://www.rapereliefshelter.bc.ca/faq/guilty.html&lt;/ref&gt; The punishment for white collar crimes are conditional sentencing, and that there is no set punishment for this particular crime.&lt;ref name=&quot;punishment white collar&quot;&gt;http://www.damianpenny.com/archived/007811.html&lt;/ref&gt; The punishment for treason is automatic life imprisonment, in Canada there is High Treason and Treason but either crime end in the punishment of life in prison.&lt;ref name=&quot;treason&quot;&gt;http://en.wikipedia.org/wiki/Treason#Canada&lt;/ref&gt; Also in Canada if you commit the crime of theft you are given a sentence of less than two years in prison and fines less than $5,000 &lt;ref name=&quot;theft&quot;&gt;http://www.lawyers.ca/statutes/criminal_code_of_canada_theft.htm&lt;/ref&gt;

Capital Punishment was abolished in Canada so that is no longer an option for sentence. For serious crimes such as murder there is a sentence of jail time, but no matter what degree of murder is you still have a chance of parole after so many determined years.&lt;ref name=&quot;canadaonline2&quot;&gt;http://canadaonline.about.com/od/crime/a/abolitioncappun.htm&lt;/ref&gt; In 1976 the option of Capital Punishment was removed from the Canadian Criminal Code. It was replaced with mandatory life sentences. In 1998 the Canadian Military removed Capital Punishment for the Canadian Nation defense Act.&lt;ref name=&quot;cap punishment&quot;&gt;http/://canadaonline.about.com/cs/crime/a/cappuntimeline.htm&lt;/ref&gt;

Prison conditions in Canada are getting worse every year. Many facilities are overcrowded and now with growing rates of mandatory sentencing the prisons are going to be even more overcrowded. It is estimated that, because of the mandatory crime rates now set in place, the Canadian Government will be spending and extra $80 million on prisons alone every year. The once good resources available to inmates to further their education and better themselves through rehab are now very limited resources, the waiting list had become so long that man inmates choose not to even enroll themselves in these programs. Canada's new &quot;tough on crime&quot; act is now making the prisoners even worse then when they first arrived to the prisons, they have a 40% re-conviction rate within two years after leaving prison. The people of Canada wants the government to be tough on crime but doesn't want to see their taxes rise along with the conviction rate.&lt;ref name=&quot;prison&quot;&gt;http://thestar.com/specialsections/crime.articles/460751&lt;/ref&gt;

In Canada you are labeled within two categories. You are either white or you are non-white. Statistics rase questions about the fairness of the Criminal Justice System in Canada. The statistics show that minorities are more likely to be warned, have DNA samples taken and more likely to be stopped by a police officer. &lt;ref name=&quot;minorities&quot;&gt;http://thestar.com/specialsections/crime.articles/460755&lt;/ref&gt; Along with minorities there are the juveniles of Canada. A person of ages 12 to 17 are viewed as juveniles of the Criminal Justice System. The Youth Criminal Justice Act came into play in April 2003, this was an act placed on the community to help adolescents have stronger ties to the community. Facts show that if a adolescent has stronger ties to the community then he/she is less likely to commit a crime. This act also separated the sentencing for adults and adolescents. Adolescents are tried in different courts along with different punishments compared to adult offenders. If an adolescent commits a crime of seriousness then he/she may be tried as an adult.&lt;ref name=&quot;juvies&quot;&gt;http://www.esc-eurocrim.org/files/ch02.pdf&lt;/ref&gt;

===Legal Personnel===
Lawyers, judges and justices all play an important role in the justice system, with small exceptions that include police officers.

Lawyers apply the law, judges and justices interpret and maintain the law. Police officers, depending on the province and its legislations, may serve as quasi-prosecutors in certain cases; for example in British Columbia, any matter that pertains to the Offence Act, especially with offences that contravene the Motor Vehicle Act. 

===Law Enforcement===
In Canada in order to become a considerable candidate for a police officer there a few requirements that need to be followed. The candidate must be 18 years old and a Canadian citizen, there is no official requirement for a college degree but it is helpful in your chances. The candidate must be certified in First Aid and CPR, you also have to pass a series of health and psychological tests. Once these and additional requirements are met, you can proceed to the application stage and compete for a position. Once selected, one is enrolled into a Police academy for further training.&lt;ref name=&quot;police officer&quot;&gt;http://www.ehow.com/how_4450748_become-police-officer-canada.html&lt;/ref&gt; Canada would be labelled as a decentralized multiple-coordinated police structure. Canada has police forces at the municipal, provincial, aboriginal federal levels. Each level of police units adheres to its own level of government and to the Canadian Criminal Code. &lt;ref name=&quot;CCJ&quot;&gt;Reichel, Philip L, Comparative Criminal Justice systems, 210&lt;/ref&gt;

Corruption in Canada is moderate. Canada is ranked 14 out of 160 countries in the world.&lt;ref name=&quot;corruption&quot;&gt;http://www.nationmaster.com/graph/gov_cor-government-corruption&lt;/ref&gt;

===Crime Rates and Public Opinion===
Canada shows a decrease in overall crime rate, it is the lowest that it has been for the past 25 years. Murder has decreased overall in Canada, it is 1.85 per 100,000 people. In the Western Provinces there seems to be a higher rate of murder compared to the Eastern Provinces. The province with the highest rate would be Saskatchewan, which has a 4.1 rate per 100,000 people. The violent crimes in Canada includes attempted murder, simple assault and assault. One out of every eight crimes there is a firearm used. Drug crimes in Canada have risen 2%. Cocaine seems to be the leading factor at 67%, Crystal Meth rose 8%, and Cannabis actually made up 60% of the entire drug offense statistics.&lt;ref name=&quot;canadaonline&quot;&gt;http://www.canadaonline.about.com/od/crime/a/crimerates2006.htm&lt;/ref&gt; The homicide rate in Canada was 594 people in the year 2007, that was 12 less then the year before. Stabbings accounted for 1/3 of it, firearms 1/3, handguns were 2/3. Police also state that one out of every 5 were gang related.&lt;ref name=&quot;the daily&quot;&gt;http://www.statcan.gc.ca/daily-quotidien/081023/dq081023a-eng.htm&lt;/ref&gt;

Public Opinion in Canada is that the punishments for crimes aren't severe enough. The people of Canada want the sentence to play a type of deterrence, protect them from the offender and reflect the seriousness of the crime in the actual sentence. A person is to serve one third of their sentence they are allowed parole, Canadians are fond of this if it is a non-violent crime. One the one third of the sentence is served then the inmate is released into a community where they are to serve the rest of the sentence under parole. Parole needs to be more strict according to 65% of Canadians, that way there is no repeat of the crime that was previously committed..&lt;ref name=&quot;public perception&quot;&gt;http://www.justice.gc.ca/eng/pi/rs/rep-rap/2001/rr01_1/p4_1.html&lt;/ref&gt;

==Rights==

===Family Law===
Canada's laws on marriage are very simple law to follow, the two people applying for their marriage license must have a valid ID along with the full name of their parents,  and the birth place of their parents. If an individual is getting re-married then their needs to be proof of a divorce or annulment, if an individual was once married and their partner is deceased then they need the death certificate. The fees for getting married vary on the Marriage License Issuer. People that allowed to get married in Canada's eyes are same sex couples, cousins and male and female couples ( over the age of 18).&lt;ref name=&quot;marriage&quot;&gt;http://marriage.about.com/cs/marriagelicenses/p/canada.htm&lt;/ref&gt; 

After marriage is an individual is no longer happy they can file for divorce. The person filing for divorce can choose to settle in court or out of court. Out side of the court room the two people getting a divorce have lawyers divide and come to an agreement on assets and child custody. If the divorce is settled in court the person after filing for divorce must submit a financial statement, that list the income of the person and the expenses and money needed to live off. Along with the financial statement can be a request for child custody. After the paperwork is submitted the papers are served to the spouse or the spouse's lawyers, they have thirty days to respond to the statement, if they disagree with the list of needs you have then it goes to a conference. If the two people cannot come to an agreement on financial needs and child custody then it moves on to a trial. The judge decides on what is best for the couple divorcing and the children (if any) involved also.&lt;ref name=&quot;divorce&quot;&gt;http://www.canadiandivorcelaws.com/divorce-procedure/&lt;/ref&gt;

Adoption is Canada is dependent on which province you live in. Each province has its own set of laws regarding adoption. Even though each province has their own set of laws there is a minimum of four home visits with a social worker and if the parents are adopting with aid from the province and not out of their own pocket then they are required to go to a PRIDE class for nine weeks. The process for adoption is very lengthy and the quickest way to adopt would to go through an agency, but it can be very expensive when doing so.&lt;ref name=&quot;adoption&gt;http://www.adoptioninformation.com/Adoption_for_Canadians&lt;/ref&gt;

===Social Inequality===

Canadians believe that the biggest social inequality is income inequality and the second; access to health care. Depending on what Province an individual lives the social inequality may vary. Studies show that the country is evenly split over this matter. Half believe there is presently greater inequality, and half believe that inequalities have decreased. People who believe that there is inequality are those belonging to the lower earning brackets in Canada, and people who don't see the inequalities belong to families with household income exceeding $80,000 a year. &lt;ref name=&quot;inequality&quot;&gt;http://www.legermarketing.com/documents/SPCLM/040802ENG.pdf&lt;/ref&gt;

===Human Rights===
Canada has it own &quot;Canadian Charter of rights and freedoms&quot; which states what rights Canadians and freedoms they may enjoy. These rights included in the Charter are the fundamental freedoms of speech, thought and association. After the fundamental rights it lists the democratic rights(the right to vote), mobility rights(come and go as you please), legal rights, equality rights (no discrimination under the law) and linguistic rights. This Charter was created by the Canadian government in 1982 to ensure the rights of their citizens.&lt;ref name=&quot;rights&quot;&gt;http://www.unac.org/rights/actguide/canada.html&lt;/ref&gt;

Every citizen in Canada is guaranteed equality. No matter what province or territory, the Federal Government will protect from discrimination.  A Canadian Citizen is protected from discrimination based on the following grounds: origin, race, creed, religion, sexual orientation, mental ability and age. The government will protect its citizens in any province where it conducts business in: hospitals, schools, human resource centers and government agencies. The Canadian government goes through numerous checks with restaurants and retail stores to ensure that their citizens' basic rights are not being violated. In Canada every class or category of people are equally protected. Every citizen is guaranteed the same right as everyone else.&lt;ref name=&quot;rights&quot;&gt;http://www.unac.org/rights/actguide/canada.html&lt;/ref&gt;

==Works Cited==
&lt;references /&gt;


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== Basic Information ==

Pakistan is a country located in Southern Asia, bordering the Arabian Sea, between India on the east and Iran and Afghanistan on the west and China in the north. The name &quot;Pakistan&quot; is came from two Persian words &quot;Pak&quot; meaning pure and &quot;stan&quot; meaning country.&lt;ref&gt;The Islamic Republic of Pakistan website accessed 10/07/2009 http://www-ec.njit.edu/~axz6893/pakistan.htm#intro&lt;/ref&gt;Pakistan covers 796,095 sq.km with a population of 176,242,949 according to population census July 2009. It is divided into four provinces: 

  
*Sindh,   
*Punjab,   
*North West Frontier Province   
*Balochistan


Climatically, Pakistan enjoys a considerable measure of variety. North and north western high mountainous ranges are extremely cold in winter while the summer months of April to September are very pleasant. The country has an agricultural economy with a network of canals irrigating a major part of its cultivated land, Wheat, cotton, rice, and sugar cane are the major crops.   Among fruits: mangos, oranges, bananas and apples are grown in abundance in different parts of the country. The main natural resources are natural gas, coal, salt and iron. The country has an expanding industry. Cotton, Textiles, sugar, cement, and chemicals play an important role in its economy.&lt;ref&gt;http://www.pakistan.gov.pk/&lt;/ref&gt; National Hazards are frequent earthquakes, occasionally severe especially in north and west; flooding along the Indus after heavy rains (July and August).&lt;ref&gt; The world factbook &quot;pakistan&quot; website accessed 10/08/2009 https://www.cia.gov/library/publications/the-world-factbook/geos/pk.html&lt;/ref&gt;

==Population==

==Brief History==

Pakistan was one of the two original successor states to British India. In August 1947, Pakistan was faced with a number of problems, some immediate but others long term. The most important of these concerns was the role played by Islam. The territory of Pakistan was divided into two parts at independence, separated by about 1,600 kilometers of Indian territory. The 1940 Lahore Resolution had called for independent &quot;states&quot; in the northwest and the northeast. This objective was changed, by a 1946 meeting of Muslim League legislators to a call for a single state. Pakistan lacked the machinery, personnel, and equipment for a new government. Even its capital, Karachi, was a second choice Lahore was rejected because it was too close to the Indian border. Pakistan's economy seemed enviable after severing ties with India, the major market for its commodities. And much of Punjab's electricity was imported from Indian power stations.&lt;ref&gt;Library of Congress Country Studies &quot;Pakistan&quot; website accessed 10/08/2009 http://countrystudies.us/pakistan/14.htm&lt;/ref&gt;

==Economic Development, Health, and Education==

==Governance==


===Elections===


===Judicial Review===


==Courts and Criminal Law==

===Punishment===


===Legal Personnel===


===Law Enforcement===

===Crime Rates and Public Opinion===
{| align=&quot;right&quot; class=&quot;wikitable&quot;
|+ '''Rates of Key Crimes in '''
| bgcolor=&quot;#E4EDEE&quot; | Lie
| bgcolor=&quot;#E4EDEE&quot; | Theft
| bgcolor=&quot;#E4EDEE&quot; | Breach of Contract
| bgcolor=&quot;#E4EDEE&quot; | Assualt
| bgcolor=&quot;#E4EDEE&quot; | Cheat
|-
|
|
|
|
|
|}

==Rights==

===Family Law===

===Social Inequality===

===Human Rights===

==References==


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{{Comparative law and justice project|region=North America}}
[[User:Blackyylubi|Blackyylubi]] 19:34, 9 February 2010 (UTC)

== Basic Information ==
A very important symbol of the country is the flag. However, the flag of the country has a very important meaning.The centered white cross that extends to the edges divides the flag into four rectangles - the top ones are blue (hoist side) and red, and the bottom ones are red (hoist side) and blue; a small coat of arms featuring a shield supported by an olive branch (left) and a palm branch (right) is at the center of the cross; above the shield a blue ribbon displays the motto, DIOS, PATRIA, LIBERTAD (God, Fatherland, Liberty), and below the shield, REPUBLICA DOMINICANA appears on a red ribbon.
&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Mar. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;The Dominican Republic is located in the Caribbean and its location is about two thirds of the island Hispaniola. Dominican Republic is in between the Caribbean Sea and the North Atlantic Ocean east of Haiti. The temperature in the Dominican Republic has seasonal variation and seasonal rainfalls. The majority of the time weather conditions compared to that of New England are wonderful.&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Mar. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;The country population is 9,650,054 as of July 2009. 31.4 percent of the people in the country are between the ages of 0-14 years old [males: 1,543,141] [females: 1,488, 016]. 62.7% are between the ages of 15-64 years old (male: 3,087,351] [females: 2,960,319]. 5.9 percent of the population is over 65 years of age [males: 2, 64,476] [females: 305,751]. This statistical information is the structure of age in the Dominican Republic since data was last recorded in 2009. The countries population has grown over 1.489 percent.
&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Mar. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;

Most of the people in the county claim to be a part of the Roman Catholic religion. About 95 present of the people in the country are Catholics. The other 5% are within some small Protestant, Seventh Day Adventist, Baptist, Mormon and Jewish communities throughout the Dominican Republic as well. There are also some people who believe in Santeria (witch craft). 
&lt;ref&gt; &quot;Dominican Republic Religion by Hispaniola.com.&quot; Dominican Republic Travel Guide. Web. Mar. 2010. &lt;http://www.hispaniola.com/dominican_republic/info/society_religion.php&gt;.&lt;/ref&gt;

[[File:Flag_of_the_Dominican_Republic.svg‎|thumb|alt=Alt text|Caption]]

==Brief History==

Hispaniola, the island on which the Dominican Republic is located, was discovered by Christopher Columbus in 1492. This happened during Christopher Columbus voyage. He decided to name it La Española. Then in 1697, Spain discovered that France had control of their neighboring country Haiti. However, Haiti is known as the third Part of the Island. The Remainder of the Island was known as Santo Domingo it was governed by Haiti for 22 years. Then in 1844 the Dominican Republic they declared independent but, in 1861 they decided to go back to the Spanish Empire. Even though after, being govern by the Spanish Empire for two years they decided to declare a war and gain back their complete independence in 1965. Dominican Republic was ran by a dictator Rafael Leonidas TRUJILLO from 1930-1961. After that Juan BOSCH was elected president in 1962 but was deposed in a military coup in 1963. However, in 1666 Juaquin Balaguer won the election and was in power and had good control over the country for 30 years. Since then, Former President (1996-2000) Leonel FERNANDEZ Reyna won election to a second term in 2004 following a constitutional amendment allowing presidents to serve more than one term.
&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Feb. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;

The Dominican Republic became independent February 27, 1844 from Haiti. However, in the country February 27 is known as a national holiday. The constitution was established November 28, 1966 and it was amended on July 25, 2002.&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Feb. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;

==Economic development, Health &amp; education==
When it comes to the birthrate within the country 22.39 /birth/   1000 in a population and the death rate is 5.28 death / 1000 in a population another, important statistic is the infant mortality rate there are 25.96 deaths within 1000 lives. The majority of the people in the country are expected to live till at least 73.7 years old. Also in the Dominican Republic there are women whom have 2 children within the period of the same year. 
When it comes to HIV/AIDS there are a lot of people in the Dominican Republic living with it. As sad as it may sound there are 62,000 throughout the country living with HIV/AIDS. Even though there have been 4,100 people in that country throughout many years that have died from HIV/AIDS.
Along with other complications there are a lot of major infections and diseases within the Dominican Republic. People at times get sick from different foods and the contaminated water or from waterborne diseases. However, people with contaminated water diseases end up having Leptospirosis. Some of those diseases are known as diarrhea, hepatitis A, Typhoid fever. The vector borne diseases are known as Malaria and dengue fever.

When it comes to health related issues the Ministry of Public health went around the country immunizing people . They immunized more than 3, 00,000 around different provinces. Their goal was to reduce the risk of different diseases.  They were also trying to reduce tetanus hepatitis b and pertussis. The country considered this to be an important event because since it is a 3rd world country and they need every health related resource that they can get. The doctor who proctored the whole thing found this to be a great thing found this to be a great opportunity for the country.

When it comes to the education in the Dominican Republic there are both public and private education institutions available. When children are in the public education system they are provided with one year of pre-schooling and with an education from grade 1-12. However, children are obligated under the laws of the country to at least go to school from pre-k till the 8th grade in order to be able to drop out. The major issue with some kids not going to school sometimes is because the education is free in public schools but, it is mandatory to wear a uniform. Since, they must wear a uniform some parents don’t have the money to pay for the uniforms and there not free. In the other hand private schools are not free. The parents of these children most pay if they want their child in private schools. Since, the parents pay for the education of their children those kids can start school at a nursery age.  When it comes to the continuation of the education after the basic 12 years of schooling there are many public and private colleges  in the country and there not free even if their public. 87% of the country population knows how to read and write. Even for those who are not fully educated by the age of 15 you are expected to know how to read and write.

The economy in the country has been established and it has been quite successful due to the whole service sector the country has.  The large tourism industry through the country has helped the country’s economy maintain stable. The free zone industry they maintain a light manufacturing through the country has also helped out the country’s economy. The most important thing that has helped the economy is the trading system of imports and exports to and from the Dominican Republic. The country is involved in major trading of exporting goods with Canada the United States and U.K.  A lot of the things that are exported from the country are: sugar, coffee, fruits, tobacco, cigars and things like that. There are goods that are imported from the United States, Venezuela and Mexico. The important things that get imported to the country are: wood, petroleum, pharmaceuticals and food products and other important goods. The whole idea of the trading industry is what has helped the countries economy get better it’s like running a business to and from the country.
However, the country suffers from a major income inequality. The poorest population receive less than one fifth of the countries (GDP).when it comes up to the rich population of the country they enjoy 40% of it. There are a high number of people in the country that do not work. Due to the recession in the United States in 2009 the counties (GDP) went down but, this year their expecting an increase. The countries rate of (GDP) is at $44.72 billion and the (GDP) in general is at $78.89 billion. But, in 2009 the rate of its increase was at a -0.3%. The money however is all equivalent in U.S dollars not in Dominican Republic money.

&lt;ref&gt;PAHO/WHO - OPS/OMS. Web. Apr. 2010. &lt;http://www.paho.org&gt;.&lt;/ref&gt;
&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Apr. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;
&lt;ref&gt;&quot;Dominican Republic Education by Hispaniola.com.&quot; Dominican Republic Travel Guide. Web. Mar. 2010. &lt;http://www.hispaniola.com/dominican_republic/info/society_education.php&gt;.&lt;/ref&gt;
&lt;ref&gt;&quot;Dominican Republic Economy by Hispaniola.com.&quot; Dominican Republic Travel Guide. Web. Mar. 2010. &lt;http://www.hispaniola.com/dominican_republic/info/society_economy.php&gt;.&lt;/ref&gt;

==Governance==
The Dominican Republic has three branches of government. In the executive branch, the chief of state is the former president Leonel Fernandez Reyna and the vice president Rafael Albuquerque de Castro ever since August 16, 2004. In other words, the president is the chief of state and the head of the government as well. In that country presidential elections are held on the same ticket as vice president. They are elected by the majority of popular vote. when they’re running in office they serve a four year term and also have the right to run for a second conservative term , the last election was held may 16 2008 and the next election will be held may 2012.From the last election Leonel Fernandez was reelected president by 53.6 percent . He ran against Miguel Vargas whose vote had 41 percent and Amable Aristy who had less than 5 percent.

When it comes to legislative ranch the cogresso nacional in other words the national congress consist of the senate known as the senado in this country there are 32 members in the senate who were elected by popular vote within the country .the house of representatives known as the camara de diputados has 178 members served for them who also are elected by popular vote .people in the senate and in the house get to serve a term of four years.

When it comes to the judicial branch the Supreme Court known as la corte suprema comprises of the president. Judges are appointed by the national judicial council as well as the president of the Supreme Court and a non congressional representative that is not governing in any way as well.&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Mar. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;

===Elections===
In Dominican Republic the election age is 18. People are allowed to vote at 18, and are also legally to participate in any government activity. People are required to vote, but not all citizens attend elections due to the fact that they feel as thouh the country does not have any improvement.Members of the arm forces and national police cannot vote. Dominicans are required to have and ID, when they take out the ID they are imediately registered for elections.They are assing a place where they are required to go vote on election date. Elections are held every four years. Presidents elections are held every four years.Two years after the presidential elections, congretional elections are held. Congretional elections consists in electing new senators, majors, and helpers of office.&lt;ref&gt;Personal Reference&lt;/ref&gt;
.&lt;ref&gt;&quot;CIA - The World Factbook.&quot; Welcome to the CIA Web Site — Central Intelligence Agency. Web. Feb. 2010. &lt;https://www.cia.gov/library/publications/the-world-factbook/geos/dr.html&gt;.&lt;/ref&gt;

==Courts and Criminal Law==

===Punishment===
In the Dominican Republic death sentences are not allowed under any circumstances. When a person murderess someone they can do up to life in prison depending on the murder.When people steal something from someone or someone’s house and there able to escape but then get caught by authority they get put into to jail and must pay a fine in order to get out of jail. However when they don’t have the money they must do the time for their crime. The weird thing about the Dominican republic in regardless to robbery is that if someone goes inside your home to try to steal from you and your at home while this act is occurring and you fear them and you shoot them for self defense and they die you don’t not get put in jail for the crime because they should of never been stealing from you in the first place.Corporal punishment under no circumstances is allowed. They don’t execute people. Since the Dominican Republic is considered as a third world country and poor compare to the United States. With this being said what talks in the Dominican Republic is money if you can’t afford a layer to back you up in a case you’re pretty much on your own and left in jail. Jail cells in the Dominican Republic are overcrowded since the prison rate is high. Two of the most known jails are La Victoria and Najayo in the capital of the Dominican Republic.In this country regardless of your age if you do a crime you must do the time. But they have a training school base jail for minors. When it comes to religious beliefs in the Dominican Republic you have the right to belief whatever you want to belief without any interference .they have bigger problems to deal with in regards’ to crime and  religion is not one of them.Citation: &lt;ref&gt;(Personal references for this information)&lt;/ref&gt;

===Legal Personnel===
When it comes to the Dominican Republic and the courts they have no jury. Judges are the ones who decide and declare a case in the aspect of weather someone is guilty or not. When it comes to hearing cases in the court room the only person to be present in the court room are the 2 lawyer from the prosecutor or the defender. In addition to that in the Dominican Republic they have the Supreme Court of justice and tribunal. When it comes to decisions of courts the Supreme Court plays the higher role in decisions and whatever the Supreme Court decides no one can object. In addition courts in the Dominican Republic run differently and are viewed differently than courts in the United States. Courts over there are divided from minor cases to major cases. It’s not like in the United States that you can have all different kind of cases going on all at once in one court room. In the Dominican Republic they like to have cases separated from each other so that all cases relating to one topic are in one room. For example all drug cases are heard in on room, and all divorce cases are heard in another. However that doesn’t mean that a judge that is only good for that one type of case it’s just the way the Dominican republic keeps there courts organized.Citation: &lt;ref&gt;(Personal references for this information)&lt;/ref&gt;

===Law Enforcement===
In the Dominican Republic they have three types of law enforcement agencies. They are called Policia Nacional Dominicana,witch is directed by Mayor General Rafael Guerrero Peralta, P.N.Direccion Nacional de Control de Drogas (DNCD) president of this department is Lic. Rolando E. Rosado Mateo. Departamento Nacional de Investigacion (DNI)is under the direction of General Ramón Antonio Aquino Garcia. Policia Nacional Dominicana is in charged of the human resources, legal process, citizens complaints and all other legal problems. Direccion Nacional de Control de Drogas is an institution whose objective is to suppress and prevent drug trafficking and drug use through preventive programs carried out at different levels of the population.Departamento Nacional de Investigaciones (DNI)is in charge of maintining national security, stability and order, as well as prosperity continuity of the Dominican State.The military is only use or needed in a case of national security. They do not attend to everyday problems, they only attend national problems and are only supposed to be on duty when the President makes the call. (DNCD) &lt;ref&gt;Policía Nacional Dominicana. Web. Mar. 2010. &lt;http://www.policianacional.gob.do/html/system/contenido.php?id_cat=4&gt;.&lt;/ref&gt;&lt;ref&gt;&quot;Direccion Nacional De Control De Drogas DNCD. Gobierno Dominicano De República Dominicana. Livio.com Tu Portal Dominicano.&quot; Directorio De República Dominicana. Livio.com Tu Portal Dominicano. Web. Mar. 2010. &lt;http://republicadominicana.livio.com/index.php?cid=16&amp;sitio=4553&gt;.&lt;/ref&gt;

===Crime Rates and Public Opinion===
When it comes to crime rates within the country in 2007 38% of deaths were in relation to crime related activities. Even though, within the time period of 2008-2009 the number of death related to crime related activities increased by 65.5%.  When it comes to the victims of these crimes 58% of them were between the age of 18 and 34 years old which is still quite young. In the majority of crimes in the country 62% happen between the hours of 6-pm till 6am and 52.5% of them happen Monday – Thursday. After all 92.6% of the victims of the crime are men that do not cooperate with the person committing the crime.&lt;ref&gt;&quot;OSAC - Dominican Republic 2009 Crime &amp; Safety Report.&quot; OSAC - Overseas Security Advisory Council. Web. Apr. 2010. &lt;http://www.osac.gov/Reports/report.cfm?contentID=95605&gt;.&lt;/ref&gt;
The most common crimes that take place in the country, are robberies and car theft as well as going into private homes to steal. Another, crime that is out of control  is the whole idea of kidnapping people for money as well as stealing peoples purses and cell phones on busy streets and in bad neighborhoods . Most of the crimes take place in the capital of the country Santo Domingo, La Vega, San José de Ocoa, San Cristobal and in Santiago. these crimes tend to be done be people who are less fortunate and are a bit poor and at times just migrate the upper class neighborhoods to steal and make a living. One of the major reasons why things like this happen in this country is because of the lack of unemployment.&lt;ref&gt;&quot;OSAC - Dominican Republic 2009 Crime &amp; Safety Report.&quot; OSAC - Overseas Security Advisory Council. Web. Apr. 2010. &lt;http://www.osac.gov/Reports/report.cfm?contentID=95605&gt;.&lt;/ref&gt; Another problem is that the country faces is their high demand with drugs and alcohol being at such an easy access for people. Since, drugs and alcohol are at such an easy access for those people who drink and are on drugs have a great possibility of getting involved in crime related activities. The whole idea of drug trading between people in the country and drug traffickers also creates problems that involve crime related activities. In addition, weapons are at a very easy access and that is one of the biggest reasons why there are a lot of crime related issues within the country.&lt;ref&gt;&quot;OSAC - Dominican Republic 2009 Crime &amp; Safety Report.&quot; OSAC - Overseas Security Advisory Council. Web. Apr. 2010. &lt;http://www.osac.gov/Reports/report.cfm?contentID=95605&gt;.&lt;/ref&gt;

==Rights==

===Family Law===
Every country has its own way to look at marriage but when you’re getting married in the Dominican Republic and you’re from the U.S you must plan accordingly with your time. Since there are a few steps you must follow before you can get married. The couple must be registered in the (city clerk) La officina Del Estado Civil.  They most both have legal passports if they’re not residents of the Dominican Republic along with an original birth certificate for each person is required. Along with that they must have an affidavit of a Single status if their single if the couple is not originally from the Dominican Republic they must pay to have it translated to Spanish by the consulate. If you happened to be getting remarried they need a certificate of divorce and under the special divorce law this person must wait 24 hours after they give in the certificate in order to get married and if they are not from Dominican Republic they must get the certificate translated in Spanish by the consulate as well. In addition, when getting married there you also need a statutory of declaration form to be notarized by a lawyer or a solicitator and each individual getting married needs one. As we all know paper work is done for free this may cost you up to $455.00 for the process of the marriage.

However if the wedding is hosted by a resort or a hotel fees may vary and they help you get the whole process complete. When getting married you must have two witnesses along with the person performing the marriage. The person whom is playing the role of witness must have an identification if their Dominican they must a (cedula) known as an ID and if the witness are not from the country they must have a valid passport. When the marriages take place as a civil ceremony after the ceremony is performed the couple is provided with documentation stating that their married and it is legal all around the world after the Oficialia Del Estado Civil will give them a marriage certificate. When you get married by the church things are a bit more formal and the couple getting married must fulfills the civil requirements as well as the church requirements. Religious marriages have the same legal rights as civil marriages when it comes to legal documentation. In the Dominican Republic men at the age of 16 and women at the age of 15 are allowed to get married without their parents consent.  No one under that age my get married unless under certain circumstances the judge allows the marriage to take place in civil marriages.  No one is allowed to be married to two people at the same time. In the case of women they must wait a time period of 10 months after the divorce became final before they can remarry unless their planning to marry the same person the divorced. However at the time that the civil marriage is taken place the government official that is hosting the ceremony has the right to waive this requirement but the wavier must be in writing. When the wedding ceremony is taken place if either the bride or the groom was married before that must provide the date and the name of the person to whom they were once married. The only way a person is no longer considered married is if one of the two spouses dies or if they get a legal divorce. The same laws apply to marriages that have taken place in the church. 


When it comes to matrimonial property laws there are different ways in which this takes place in this country you either both have rights to the property under the common property law when both men and the women have the same rights to everything owned between the couple. As well as the separate property law when everyone is left responsible of like their own debt within the marriage but the husband is left responsible for his own debt and for the debt of the wife while he was still being represented as if he was married.

On august 8, 1990 the Dominican Republic signed a convention on the rights of children. Then on June 11, 1991 the convention was ratified.  In 2004 the country adopted a new code that protected children and adolescents law163-03 El Codigo para el Sistema de Protection y los Derechos Fundamentales de Los Niños y niñas. This was related to everything in the country in regards to children for example parental rights, adoption, guardianship and custody.  Children have the right to speak up and give their opinion when it comes to the whole idea of any of these cases according to the law in the Dominican Republic. When all of this was established they came up with a family court system for children know as Tribunal del niño y niñas y adolescente.  However since resources are limited in the country sometimes laws can be ignored. Parent’s rights can be terminated if the child is being neglected and mistreated especially when there is evidence to prove it. Researches shown that the majority of children in the Dominican Republic where or are victims of abuse but because of the lack of resources lots of cases are un- reported. In most cases the person being the abuser is either a close family friend or a relative. However, they are trying to come up with a way to have these thing not happen and find a solution to the problem but it is very time consuming but their working on it. By law people should not be mistreating children they must care for them but not abuse them.&lt;ref&gt;&quot;Legal Options For Marriage in the Dominican Republic - U.S. Embassy in Santo Domingo, Dominican Republic.&quot; Embassy of the United States Santo Domingo, Dominican Republic - Home. Web. Apr. 2010. &lt;http://santodomingo.usembassy.gov/marriage_dr-e.html&gt;.&lt;/ref&gt;&lt;ref&gt;&quot;Legal Options For Divorce in the Dominican Republic - U.S. Embassy in Santo Domingo, Dominican Republic.&quot; Embassy of the United States Santo Domingo, Dominican Republic - Home. Web. Apr. 2010. &lt;http://santodomingo.usembassy.gov/divorce_dr-e.html&gt;.&lt;/ref&gt;&lt;ref&gt;&quot;Legal Requirements to Get Married in The Dominican Republic.&quot; CaribbeanWeddings.comÂ® - Your Source for Travel Information about Caribbean Weddings - Have Your Wedding in the Caribbean -. Web. Apr. 2010. &lt;http://www.caribbeanweddings.com/legal_requirements,islands,Dominican_Republic.html&gt;.&lt;/ref&gt;

===Human Rights &amp; Social inequaliy===
In Dominican Republic the number of unlawful killings by security has raised in 2008. When it comes to the Haitian Dominicans and Haitians in general they face some serious issues within the country in ideal of human rights. When it comes to a person’s rights to health they try to warn people to be advised of people who have HIV &amp; AIDS. Under the rights to a person’s health the Joint United Nation Programmer on HIV &amp; AIDS warn people that there was an epidemic in regards to AIDS. There was a point in time that there were more people with HIV &amp; AIDS than what the country was able to control. In poor neighborhoods were people of low resources lived like batays has highest demand for HIV &amp; AIDS due to their lack of resources.


People with HIV &amp; AIDS in the country have to face a very high demand of discrimination within the country in regards to the acceptance by society. When it comes to the work place people fear to hire people with AIDS they are high discriminated because of their illness. This how a human right is described as to when it comes to a serious illness in the country.When it comes to the right of policing and security the force they have a very lacking ability in their government and even in the way of helping and trying to solve a situation. They have a very high level of violent crimes in some neighborhoods. Information states there where 298 people who were killed by the police between January and August however, there has been a major concern in regards to human rights because the number of fatal shootings have been very unlawful. That led to having a lot of police officers expelled from the service for beginning corrupted and were asked to retake some training to try to improve their policing, but since not many allegations were made the system still continues to be corrupt with individuals and rights. There are millions of cases where police officers take advantage of the people’s rights just because they are an authority figure.When it comes to access of nationality they also face a high rate of discrimination because of the fact that there are a lot black of dark skin colored Dominicans. So what they do is examine closely documentation before renewal or registration. However under the human rights they should not be discriminated because of their color. They were also mandated to immediately provide legal documentation to all Dominican Haitians. Its shows that regardless of the discrimination they play against them they have human rights that protects them.


When it comes to expulsions in the country in regards to the human rights organization there were more than 6000 Haitians that were sent back to Haiti in the first 6 months of the year. The majority of those deportations that took place were not done legally right under the international legal right standard. They were reports that the state they were mistreated by immigration officials and security forces.
In addition the human right organization also stated that there were many Haitian children who happen to be trafficked into the Dominican Republic for the use of agricultural and domestic work. That was just not right the way those children were being treated as a working object instead of providing them with a fuel blown education  but organization of human rights did try to help those children who’s cases were reported. 
When it comes to the rights of freedom of expressions of human under the right of a journalist in the Dominican Republic reports have shown that many of the workers are intimidated harassed on the job as to how they are to express them self’s  on their job. At times they where even threatened by their employer physically or verbally as to the things they can or can not say.&lt;ref&gt;
&quot;Dominican Republic | Amnesty International.&quot; Amnesty International | Working to Protect Human Rights. Web. Apr. 2010. &lt;http://www.amnesty.org/en/region/dominican-republic/report-2009&gt;.&lt;/ref&gt;

===Works Cited===
&lt;references/&gt;
http://venyve.com/republica-dominicana

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== Basic Information ==
	

[[File:Flag_of_Cuba.svg‎ |thumb|alt=Alt text|Caption]]
[[File:Cu-map.png#file]]

'''Geographical Information''' Cuba is in the Caribbean. It is an island in between the Caribbean sea and the North Atlantic Ocean. The island consists of 110,860 sq km and is 150 km south of Key West, FLorida.&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; Cuba is the largest country in the Caribbean and westernmost island of the Greater Antilles. &lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; It is slightly smaller than Pennsylvania. Cuba borders the US Naval Base at Guantanamo Bay 29 km.&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; '''Demographic Characteristics''' Its population is ranked 72 in comparison to the world with a total population of 11,451,652 (July 2009 est.)&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; The age structure in Cuba consists of 0-14 years: 18.3% (male 1,077,745/female 1,020,393)15-64 years: 70.4% (male 4,035,691/female 4,030,103)65 years and over: 11.2% (male 584,478/female 703,242) (2009 est.) The median age is a total of: 37.3 years and 36.6 years for males and 38 years for females. (2009 est.) '''Religion''' Cuba is 85% Roman Catholic prior to CASTRO assuming power; Protestants, Jehovah's Witnesses, Jews, and Santeria are also represented. &lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; '''Ethnic Groups''' According to the 2002 census, there are: 65.1% white, 24.8% mulatto and mestizo, 10.1% black. '''Language''' Spanish is widely spoken in Cuba.

==Brief History==
In 1492, Christopher Columbus made the European discovery of the Island of Cuba. &lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; Spanish settlers established the raising of cattle, sugarcane, and tobacco as Cuba's primary economic pursuits.&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt;  As the native Indian population died out, African slaves were imported to work the ranches and plantations. Slavery was abolished in 1886.&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; 

Cuba was the last major Spanish colony to gain independence, following a lengthy struggle begun in 1868. &lt;ref&gt;Department Of the State&lt;/ref&gt;Jose Marti, Cuba's national hero, helped initiate the final push for independence in 1895. In 1898, the United States entered the conflict after the USS Maine sank in Havana Harbor on February 15 due to an explosion of undetermined origin.&lt;ref&gt;Department Of the State&lt;/ref&gt;In December of that year, Spain relinquished control of Cuba to the United States with the Treaty of Paris. On May 20, 1902, the United States granted Cuba its independence but retained the right to intervene to preserve Cuban independence and stability in accordance with the Platt Amendment. &lt;ref&gt;Department Of the State&lt;/ref&gt;In 1934, the Platt Amendment was repealed. The United States and Cuba concluded a Treaty of Relations in 1934 which, among other things, continued the 1903 agreements that leased the Guantanamo Bay naval base to the United States. &lt;ref&gt;Department Of the State&lt;/ref&gt;

==Economic Development, Health, and Education==

'''Economic Development''' The loss of the Soviet aid and domestic inefficiencies has caused for the average Cuban's standard of living to be at a the lower level. Since 2000, Venezuela has been providing oil on preferential terms, and it currently supplies about 100,000 barrels per day of petroleum products. Cuba has been paying for the oil, in part, with the services of Cuban personnel in Venezuela including some 30,000 medical professionals.&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; The GDP (purchasing power parity) is $110.8 billion (2009 est.) Cuba's country comparison to the world is ranked at 65. The GDP per capita (PPP) is $9,700 (2009 est.) and it's country comparison to the world is ranked at 109. Cuba's industries consist of sugar, petroleum, tobacco, construction, nickel, steel, cement, agricultural machinery, pharmaceuticals. Cuba's imports are estimated to be at $10.86 billion (2009 est.) with a country comparison to the world ranked at 83.&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; Import commodities are petroleum, food, machinery and equipment, chemicals. With import partners being Venezuela 29.8%, China 11.8%, Spain 10%, Canada 6.4%, US 6.3%, Brazil 4.6% (2008).&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; Cuba's exports are estimated to be at $3.253 billion (2009 est.) with a country comparison to the world ranked at 118. It's export commodities are sugar, nickel, tobacco, fish, medical products, citrus, coffee. With export partners being Canada 27.8%, China 26.7%, Spain 6.2%, Netherlands 5.6%(2008).&lt;ref&gt;CIA https://www.cia.gov/library/publications/the-world-factbook/geos/cu.html&lt;/ref&gt; '''Health''' After the 1959 revolution, health conditions have greatly improved and sanitation is generally really good. On the other hand, Cuba no longer receives the same level of foreign support and has fallen behind in many of its social services due to the dissolution of the Soviet Union. in 1993 100% of the population was reported to have access to health care. In 2000, 95% of the population had access to safe drinking water and 95% had adequate sanitation. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt;

Infant mortality declined from more than 60 per 1,000 live births before 1959 to 6 in 2000. About 8% of babies born in 1999 were considered low birth weight.&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; Studies show that as of 1998 79% of married women (ages 15 to 49) used contraception.&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; The government claims to have eradicated malaria, diphtheria, poliomyelitis, tuberculosis, and tetanus. In 1997, children up to one year of age were immunized as follows: tuberculosis, 99%; diphtheria, pertussis, and tetanus, 99%; polio, 97%; and measles, 99%. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt;

Life expectancy was an average of 76 years for women and men in 2000 and the overall death rate was 7 per 1,000 people in 1999.&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; In 2000, major causes of death, with incidence per 100,000 people, were circulatory system diseases (235), cancer (124), injuries (61), and infectious diseases (44).&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; There were 15 reported cases of tuberculosis per 100,000 in 1999. The incidence of TB decreased steadily from 1979 to 1991 (503 cases), but there was a reversal in the following years.&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; As of 2001 the number of people living with HIV/AIDS was estimated at 2,800, and deaths from AIDS that year were estimated at 120. HIV prevalence was 0.06 per 100 adults. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt;

'''Education''' Education has been a high priority of the Castro government. In 1959 there were at least one million illiterates, and many more were only semi literate. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; A literacy campaign was inaugurated in 1961, when 100,000 teachers went out into the countryside. For the year 2000, UNESCO estimated the illiteracy rate of persons aged 15 years and over to be 3.6% (males, 3.5%; females, 3.6%).&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; Education is free and compulsory for six years (6–11 years of age).&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; In 1997, Cuba's 9,926 primary schools enrolled 1,094,868 students and employed 92,820 teachers. Student-to teacher ratio stood at 12 to 1. Secondary education lasts six years beyond the primary level. In 1997, 712,897 secondary students were instructed by 70,628 teachers.&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; As of 1999, 99% of primary-school-age children were enrolled in school, while 80% of those eligible attended secondary school. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt;The pupil-teacher ratio at the primary level was 12 to 1 in the same year. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; Cuba has five universities: the University of Havana (founded 1728), Oriente University at Santiago de Cuba (1947), the University of Las Villas at Santa Clara (1952), University of Camagüey (1974), and the University of Pinar Del-Rio. Workers' improvement courses (superación obrera), to raise adults to the sixth-grade level, and technical training schools (mínimo técnico), to develop unskilled workers' potentials and retrain other workers for new jobs, were instituted after 1961.&lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt; Today, special worker-farmer schools prepare workers and peasants for enrollment at the universities and for skilled positions in industrial and agricultural enterprises. In 1997, tertiary education enrolled 111,587 students, with 22,574 teachers. &lt;ref&gt; Encyclopedia Of Nations&lt;/ref&gt;

==Governance==

Constitution: February 24, 1976; amended July 1992 and June 2002 

The Constitution of 1976, which defined Cuba as a socialist republic, was replaced by the Constitution of 1992, which is guided by the ideas of José Martí, Marx, Engels and Lenin.&lt;ref&gt;Wapedia&lt;/ref&gt; [4] The constitution describes the Communist Party of Cuba as the &quot;leading force of society and of the state&quot;.&lt;ref&gt;Wapedia&lt;/ref&gt; [4] The first secretary of the Communist Party, is concurrently President of the Council of State (President of Cuba) and President of the Council of Ministers (sometimes referred to as Prime Minister of Cuba).&lt;ref&gt;Wapedia&lt;/ref&gt; [74] Members of both councils are elected by the National Assembly of People's Power. &lt;ref&gt;Wapedia&lt;/ref&gt;

According to the Cuban constitution, Cuba is an independent socialist republic that is controlled by 1 party: the Cuban Communist Party (PCC), of which Fidel Castro is the head, with his brother, Raul Castro as vice-president.&lt;ref&gt; Encyclopedia of the Nations&lt;/ref&gt; The Communist Party is led by a group of 25 individuals chosen by its head. Molded by this elite group of communists are organizations that encompass every facet of society, including youth, women, workers, and small farmers, among others.&lt;ref&gt; Encyclopedia of the Nations&lt;/ref&gt; Around 80 percent of the population has membership in at least one of these organizations. This network ensures that the agenda of the Communist Party is disseminated (communicated) to the masses. The National Assembly is the legislative body of the Cuban government. &lt;ref&gt; Encyclopedia of the Nations&lt;/ref&gt;The Assembly is composed of 601 members whose terms last 5 years. For these positions, the Council of State nominates candidates, who are then subject to a direct vote by the Cuban people.&lt;ref&gt; Encyclopedia of the Nations&lt;/ref&gt; The National Assembly also elects the Judicial Branch. On the local level, members of Municipal Assemblies are chosen by direct local election. Local government is closely over-seen by the Communist Party.&lt;ref&gt; Encyclopedia of the Nations&lt;/ref&gt; As is evidenced by Fidel Castro's almost complete control over decision-making, most policies are the direct result of his personal desires. The Cuban governmental structure is heavily bureaucratic (organized into many agencies). Until 1993, the Central Planning Board (JUCEPLAN, or Junta de Planificación Central), was responsible for economic planning. After 1993, in a move to create greater efficiency and to decentralize, different sectors of the economy became the responsibility of various ministerial bodies, including the Ministry of Tourism, the Ministry of Science, Technology, and the Environment, the Ministry of Industry, the Ministry of Sugar Planning, and the Ministry of Foreign Investment and Economic Cooperation, among others. &lt;ref&gt; Encyclopedia of the Nations&lt;/ref&gt; 

'''Executive Branch''' Chief of state: President of the Council of State and President of the Council of Ministers Gen. Raul CASTRO Ruz (president since 24 February 2008)&lt;ref&gt; Countries of the World http://www.theodora.com/wfbcurrent/cuba/cuba_government.html&lt;/ref&gt;
'''Legislative Branch''' Unicameral National Assembly of People's Power or Asemblea Nacional del Poder Popular (number of seats in the National Assembly is based on population; 614 seats; members elected directly from slates approved by special candidacy commissions to serve five-year terms) )&lt;ref&gt; Countries of the World http://www.theodora.com/wfbcurrent/cuba/cuba_government.html&lt;/ref&gt;
'''Judicial Branch''' People's Supreme Court or Tribunal Supremo Popular (president, vice president, and other judges are elected by the National Assembly)&lt;ref&gt;Countries of the World http://www.theodora.com/wfbcurrent/cuba/cuba_government.html&lt;/ref&gt;
===Elections===

Elections in Cuba take place by secret ballot.&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;
Since Cuba became a one-party republic and the Communist party became the official political party, Cuba has been both condemned and praised by certain Cuban groups, international groups, and foreign governments regarding democracy. Although the media is operated under the supervision of the Communist Party’s Department of Revolutionary Orientation, which &quot;develops and coordinates propaganda strategies&quot; &lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;

==Courts and Criminal Law==

In 1973, the Cuban government promulgated a new Law of Judicial Organization. This law established a hierarchical and more formal court system, replaced the private practice of law with law collectives known as bufetes colectivos, and strengthened the emphasis on &quot;socialist legality.&quot;&lt;ref&gt; Comparative Criminology http://www-rohan.sdsu.edu/faculty/rwinslow/namerica/cuba.html&lt;/ref&gt; This period was also marked by increasingly close relations with the Soviet Union, and increased economic dependence on COMECON - the trading bloc of socialist nations. &lt;ref&gt; Comparative Criminology http://www-rohan.sdsu.edu/faculty/rwinslow/namerica/cuba.html&lt;/ref&gt; The Supreme Court of Cuba serves as the nation's highest judicial branch of government. It is also the court of last resort for all appeals against the decisions of provincial courts. &lt;ref&gt;Wapedia&lt;/ref&gt;

===Legal Personnel===

Independent legal practice is not permitted in Cuba. As of 1999, 30% of lawyers worked as legal advisors to state agencies, ministries, and commercial enterprises.&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;These lawyers receive a lower salary than their counterparts in the bufetes, but this is offset somewhat by added perks and bonuses from their employer.&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;The salary of lawyers is based upon the number and complexity of the cases which they handle. Better lawyers typically earn a higher salary.&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;Bufetes Colectivos are collective law offices, first established by the Ministry of Justice after the private practice of law was abolished, and currently under the oversight of the National Organization of Bufetes Colectivos (ONBC).&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;
In order to practice in a bufete, one must graduate from law school in Cuba or a foreign country with Cuban validation. Exceptions to this can be made under extraordinary circumstances. Once in a bufete, lawyers may practice anywhere in the country.&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;
Currently, approximately 2,000 lawyers practice in some 250 bufetes throughout Cuba, collectively handling some 200,000 cases per year.&lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt; Lawyers in bufetes typically have large caseloads and work under difficult conditions. A small number of bufetes specializing in providing legal assistance to foreign nationals have arisen in recent years. &lt;ref&gt;Wikipedia The Free Encyclopedia&lt;/ref&gt;

===Law Enforcement===

The Ministry of Interior is the principal entity of state security and totalitarian control. Officers of the Revolutionary Armed Forces (FAR), which are led by Raul Castro, the President's brother, have been assigned to the majority of key positions in the Ministry of Interior in the past several years.&lt;ref&gt; Comparative Criminology http://www-rohan.sdsu.edu/faculty/rwinslow/namerica/cuba.html&lt;/ref&gt; In addition to the routine law enforcement functions of regulating migration and controlling the Border Guard and the regular police forces, the Interior Ministry's Department of State Security investigates and actively suppresses political opposition and dissent.&lt;ref&gt; Comparative Criminology http://www-rohan.sdsu.edu/faculty/rwinslow/namerica/cuba.html&lt;/ref&gt; It maintains a pervasive system of surveillance through undercover agents, informers, rapid response brigades (RRB's), and neighborhood-based Committees for the Defense of the Revolution (CDR's). &lt;ref&gt; Comparative Criminology http://www-rohan.sdsu.edu/faculty/rwinslow/namerica/cuba.html&lt;/ref&gt;

===Crime Rates and Public Opinion===

*Executions  5 executions  [20th of 22]
*Murders committed by youths  348  [15th of 73]  
*Murders committed by youths per capita  9.6  [16th of 57] 
*Prisoners  0 prisoners  [156th of 168]
*Prisoners &gt; Per capita  0 per 100,000 people  [157th of 164]  


Cuba is principally a source country for women and children trafficked within the country for the purpose of commercial sexual exploitation and possibly for forced labor; the country is a destination for sex tourism, including child sex tourism, which is a problem in many areas of the country; some Cuban nationals willingly migrate to the United States, but are subsequently exploited for forced labor by their smugglers; Cuba is also a transit point for the smuggling of migrants from China, Sri Lanka, Bangladesh, Lebanon, and other nations to the United States and Canada&lt;ref&gt;Nation Master http://www.nationmaster.com/country/cu-cuba/cri-crime&lt;/ref&gt;

Violent crime is officially viewed as a threat to national stability. In 1999 the death penalty was extended to certain narcotics offenses, robbery involving firearms, attacks on security officers, and sexual corruption of minors. In the biggest crackdown in a decade, 75 prodemocracy activists were arrested in March 2003 and summarily tried. Shortly afterward a de facto three-year moratorium on executions was ended by the execution of three ferry hijackers.&lt;ref&gt;World Desk Reference http://dev.prenhall.com/divisions/hss/worldreference/CU/crime.html&lt;/ref&gt;


Estimated arrests for drug abuse violations by age group, 2000-2007

*Year:2000 Adult: 1,375,600 Juvenile: 203,900 
*Year:2001 Adult: 1,384,400 Juvenile: 202,500 
*Year:2002 Adult: 1,352,600 Juvenile: 186,200 
*Year:2003 Adult: 1,476,800 Juvenile: 201,400 
*Year:2004 Adult: 1,551,500 Juvenile: 194,200 
*Year:2005 Adult: 1,654,600 Juvenile: 191,800 
*Year:2006 Adult: 1,693,100 Juvenile: 196,700 
*Year:2007 Adult: 1,645,500 Juvenile: 195,700 
Source: Crime in the United States, annual, Uniform Crime Reports 


Nonfatal firearm-related violent crimes, 2000-2008 &lt;ref&gt;http://www.ojp.usdoj.gov/flash.htm&lt;/ref&gt;
                            
*Year: 2000 Firearm Incidents: 428,670 Firearm Victims: 533,470 Firearm Crime Rate: 2.4 Firearm Crimes as percent of all violent incidents: 7
*Year: 2001 Firearm Incidents: 467,880 Firearm Victims: 524,030 Firearm Crime Rate: 2.3 Firearm Crimes as percent of all violent incidents: 9
*Year: 2002 Firearm Incidents: 353,880 Firearm Victims: 430,930 Firearm Crime Rate: 1.9 Firearm Crimes as percent of all violent incidents: 7 
*Year: 2003 Firearm Incidents: 366,840 Firearm Victims: 449,150 Firearm Crime Rate: 1.9 Firearm Crimes as percent of all violent incidents: 7
*Year: 2004 Firearm Incidents: 280,890 Firearm Victims: 331,630 Firearm Crime Rate: 1.4 Firearm Crimes as percent of all violent incidents: 6              
*Year: 2005 Firearm Incidents: 416,940 Firearm Victims: 474,110 Firearm Crime Rate: 1.9 Firearm Crimes as percent of all violent incidents: 9   
*Year: 2006            
*Year: 2007 Firearm Incidents: 348,910 Firearm Victims: 394,580 Firearm Crime Rate: 1.6 Firearm Crimes as percent of all violent incidents: 7   
*Year: 2008 Firearm Incidents: 303,880 Firearm Victims: 343,550 Firearm Crime Rate: 1.4 Firearm Crimes as percent of all violent incidents: 7

==Rights==

===Family Law===

The Family Code was developed in the early 1974. The Family Code was so important to the Cuban people that they deemed it vital to have a complete and “far reaching” discussion about it. People as young as junior high school students got enthusiastically interested in the Code, and had debates and discussions about it as the first law to have tremendous importance to their future. The plan for the discussion of the code was announced by Blas Roca at the Women’s Congress. Roca was a very active member of the Orthodox party. And by then he was Secretariat and head of the committee to draft new laws. He is now the president of the national People’s Assembly. 
The Family Code covers marriage, divorce, marital property relationships, recognition of children, obligations for children’s care and education, adoption, and tutelage.&lt;ref&gt; Wikipedia The Free Encyclopedia&lt;/ref&gt; The following are Clauses 24, 25, 26, 27, and 28 of the Cuban Family Code:

24. Marriage is constituted on the basis of equal rights and duties of both partners. .&lt;ref&gt; Wikipedia The Free Encyclopedia&lt;/ref&gt;

25. The spouses must share the same home, be faithful to one another, help, consider and respect each other. The rights and duties established by this code will subsist in their entirety as long as the marriage has not been legally terminated, in spite of the fact that for justifiable reasons a common household cannot be maintained..&lt;ref&gt; Wikipedia The Free Encyclopedia&lt;/ref&gt;

26. Both spouses are obligated to care for the family they have created and cooperate with each other in the education, formation and guidance of their children in line with the principles of socialist morality. As well, each to the extent of his or her capabilities and possibilities must participate in governing the home and cooperate toward its best possible care..&lt;ref&gt; Wikipedia The Free Encyclopedia&lt;/ref&gt;

27. The spouses are obligated to contribute toward satisfying the needs of faculties and economic capacities. Nevertheless, if one of the spouses contributes only through his or her work in the home and child-care, the other spouse must provide full economic support without this meaning that he or she be relieved of the obligations of cooperating with the housework and child-care..&lt;ref&gt; Wikipedia The Free Encyclopedia&lt;/ref&gt;

28. Both spouses have the right to exercise their professions or crafts and must lend each other reciprocal cooperation and aid to this effect, as well as in order to carry out studies or perfect their training, but in all cases they will take care to organize their home life so that such activities be coordinated with fulfillment of the obligations imposed by this code.” .&lt;ref&gt; Wikipedia The Free Encyclopedia&lt;/ref&gt;

===Human Rights===

Cuba's government controls all aspects of life through the Communist Party and its affiliated mass organizations, the government bureaucracy, and State Security Department.&lt;ref&gt;Department Of State&lt;/ref&gt;The latter is tasked with monitoring, infiltrating, and controlling the country's beleaguered human rights community.&lt;ref&gt;Department Of State&lt;/ref&gt;Despite having signed the International Covenant on Civil and Political Rights and the International Covenant on Economic, Social, and Cultural Rights in February 2008, Cuba has yet to ratify either or meet the obligations assumed in these instruments, continuing to commit serious abuses and denying its citizens the right to change their government.&lt;ref&gt;Department Of State&lt;/ref&gt;Cuba is also a signatory of the Universal Declaration of Human Rights and sits on the UN Human Rights Council, yet routinely arrests citizens who seek to exercise internationally recognized fundamental freedoms.&lt;ref&gt;Department Of State&lt;/ref&gt;The government incarcerates people for their peaceful political beliefs or activities. The total number of political prisoners and detainees is unknown, because the government does not disclose such information and keeps its prisons off-limits to human rights organizations and international human rights monitors.&lt;ref&gt;Department Of State&lt;/ref&gt; One local human rights organization lists more than 200 political prisoners currently detained in Cuba in addition to as many as 5,000 people sentenced for &quot;dangerousness.&quot; &lt;ref&gt;Department Of State&lt;/ref&gt;

Cuba, with a population of approximately 11 million, is a totalitarian state that does not tolerate opposition to official policy.  &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt; The country is led by Raul Castro, who holds the positions of chief of state, president of the council of state and council of ministers, and commander in chief of the Revolutionary Armed Forces. &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt;Although the constitution recognizes the unicameral National Assembly as the supreme authority, the Communist Party (CP) is recognized in the constitution as the only legal party and &quot;the superior leading force of society and of the state.&quot; Fidel Castro remained the first secretary of the CP.  &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt;The January 2008 elections for the National Assembly were neither free nor fair, and all of the candidates had to be preapproved by a CP candidacy commission, with the result that the CP candidates and their allies won 98.7 percent of the vote and 607 of 614 seats in the National Assembly. &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt;Civilian authorities, through the Ministry of the Interior, exercised control over the police, the internal security forces, and the prison system.


The government continued to deny its citizens their basic human rights, including the right to change their government, and committed numerous and serious abuses. &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt;The following human rights problems were reported: beatings and abuse of prisoners and detainees, harsh and life-threatening prison conditions, including denial of medical care; harassment, beatings, and threats against political opponents by government‑recruited mobs, police, and state security officials acting with impunity; arbitrary arrest and detention of human rights advocates and members of independent professional organizations; and denial of fair trial, including for at least 194 political prisoners and as many as 5,000 persons who have been convicted of potential &quot;dangerousness&quot; without being charged with any specific crime. 


Authorities interfered with privacy and engaged in pervasive monitoring of private communications.  &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt;
There were also severe limitations on freedom of speech and press; denial of peaceful assembly and association; restrictions on freedom of movement, including selective denial of exit permits to citizens and the forcible removal of persons from Havana to their hometowns; and restrictions on freedom of religion and refusal to recognize domestic human rights groups or permit them to function legally. Discrimination against persons of African descent, domestic violence, underage prostitution, trafficking in persons, and severe restrictions on worker rights, including the right to form independent unions, were also problems.. &lt;ref&gt;Department Of State http://www.state.gov/g/drl/rls/hrrpt/2009/wha/136108.htm&lt;/ref&gt;


The government places severe limitations on freedom of speech and press, as noted by international non-governmental organizations (NGOs) such as Reporters Without Borders.&lt;ref&gt;Department Of State&lt;/ref&gt;The constitution provides for freedom of speech and press insofar as views &quot;conform to the aims of a socialist society.&quot;&lt;ref&gt;Department Of State&lt;/ref&gt;In March 2008, demonstrators distributing copies of the Universal Declaration of Human Rights were attacked by an orchestrated mob and later detained.&lt;ref&gt;Department Of State&lt;/ref&gt;Despite the government's decision to permit Cubans to purchase personal computers, access to the Internet is strictly controlled and given only to those deemed ideologically trustworthy; Internet restrictions were tightened further in March and April 2008 to block access by Cuban citizens to certain independent websites.&lt;ref&gt;Department Of State&lt;/ref&gt;

Freedom of assembly is not constitutionally guaranteed in Cuba.&lt;ref&gt;Department Of State&lt;/ref&gt;
The law punishes unauthorized assembly of more than three persons. The government also restricts freedom of movement and prevents some citizens from emigrating because of their political views. Cubans need explicit exit visas from their government to leave their country, and many people are denied exit permission by the Cuban Government, despite the fact that they have received travel documents issued by other countries. &lt;ref&gt;Department Of State&lt;/ref&gt;

===Works Cited===
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

Here's my [[User:Jtneill/Teaching/Philosophy|teaching philosophy]]. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

[[/Research|My research]] expertise includes [[outdoor education]], [[green exercise]], and the effects of experiential group-based intervention programs. More broadly, I am interested in [[positive psychology]] and [[environmental psychology]].

[[/Presentations|I present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

I'm currently working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|open wiki assignments for authentic learning]].

I most recently presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and permaculture gardening.

[[/Contact|Feel free to connect.]]
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My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

[[/Research|My research]] expertise includes [[outdoor education]], [[green exercise]], and the effects of experiential group-based intervention programs. More broadly, I am interested in [[positive psychology]] and [[environmental psychology]].

[[/Presentations|I present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

I'm currently working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|open wiki assignments for authentic learning]].

I most recently presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and permaculture gardening.

[[/Contact|Feel free to connect.]]
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      <comment>+ [[w:guerilla gardening|guerilla gardening]]</comment>
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

[[/Research|My research]] uses quantitative and qualitative psychological methods to examine the processes and effects of [[outdoor education]], [[adeventure therapy]], [[green exercise]], and other group-based experiential intervention programs. More broadly, I am interested in intersections of [[positive psychology]] and [[environmental psychology]].

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

[[/Research|My research]] uses quantitative and qualitative psychological methods to examine the processes and effects of [[outdoor education]], [[adventure therapy]], [[green exercise]], and other group-based experiential intervention programs. More broadly, I am interested in intersections of [[positive psychology]] and [[environmental psychology]].

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

[[/Research|My research]] uses quantitative and qualitative psychological methods to examine the processes and effects of [[outdoor education]], [[adventure therapy]], [[green exercise]], and other group-based experiential intervention programs. More broadly, I am interested in intersections of [[positive psychology]] and [[environmental psychology]].

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|Open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

{{/Research}}

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|Open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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      <id>2804468</id>
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      <comment>+ verify</comment>
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]&lt;small&gt;&lt;sup&gt;[https://en.wikiversity.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Jtneill (verify)]&lt;/sup&gt;&lt;/small&gt;. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

{{/Research}}

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|Open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]&lt;small&gt;&lt;sup&gt;[https://en.wikiversity.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Jtneill (verify)]&lt;/sup&gt;&lt;/small&gt;. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]]
* ~[https://xtools.wmcloud.org/ec/en.commons/Jtneill 2,200 edits] on [[c:|Wikimedia Commons]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I currently teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

{{/Research}}

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|Open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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&lt;div style=&quot;background:white; border:2px SteelBlue solid; padding:12px;&quot;&gt;
My name is James Neill (''he/him''). I'm an Assistant Professor in the [https://www.canberra.edu.au/about-uc/faculties/health/study/psychology Discipline of Psychology] at the [[University of Canberra]], Australia.

I am passionate about [[open academia]]—I like to share knowledge openly.

On English Wikiversity, I'm a [[WV:Custodianship|custodian]] and [[WV:Bureaucratship|bureaucrat]]&lt;small&gt;&lt;sup&gt;[https://en.wikiversity.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Jtneill (verify)]&lt;/sup&gt;&lt;/small&gt;. Since 2005, I've made:
* ~[https://xtools.wmcloud.org/ec/en.wikiversity.org/Jtneill 80,000 edits] on  [[Main page|Wikiversity]]
* ~[https://xtools.wmcloud.org/ec/en.wikipedia/Jtneill 4,600 edits] on [[w:|Wikipedia]]
* ~[https://xtools.wmcloud.org/ec/en.commons/Jtneill 2,200 edits] on [[c:|Wikimedia Commons]].

My [[User:Jtneill/Teaching/Philosophy|teaching philosophy]] is based on experiential learning. [[/Teaching|I teach]] a 3rd-year undergraduate [[psychology]] unit, [[motivation and emotion]], and a 4th-year Honours unit, [[research methods in psychology]].

{{/Research}}

[[/Presentations|I also present]] about open education, wikis in higher education, and collaborative development of open educational resources. 

Currently, I'm working on:
[[User:Jtneill/Presentations/Open wiki assignments for authentic learning|Open wiki assignments for authentic learning]].

Most recently, I presented on:
[[User:Jtneill/Presentations/Interactive classroom exercises using Google Forms and Sheets|Interactive classroom exercises using Google Forms and Sheets]].

Hobbies include exploring outdoors, mountain biking, and [[w:guerilla gardening|guerilla gardening]].

[[/Contact|Feel free to connect.]]
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      <text bytes="8651" sha1="ov22j1nlel14y4apreb8genuqztro33" xml:space="preserve">Part of the [[Comparative law and justice]] Wikiversity Project
{{Comparative law and justice project|region=Europe}}
[[File:Flag of Luxembourg.svg|border|right|200px]]


== Basic Information ==	
'''''Population '''''
*The total population in Luxembourg is 491,775 as of July 2010. The population is broken down into 46,918 male and 44,052 females, between the ages of 0-14, who make up 18.5% of the population.66.7% of the population is made up of 165,342 males and 162,681 females between the ages of 15 to 64 years old.  29,839 males and 42,943 females make up the rest of the population consists of 14.8% of people who are age 65 and older.

'''''Geographic Information'''''
*Luxembourg is located in Western Europe between the countries of France and Germany. France, Germany, and the country of Belgium boarder the country of Luxembourg. The total Land mass in Luxembourg is 2,586 sq km. Usually the climate in Luxembourg consists of mild winters and cool summers. The city of Luxembourg is the largest in the country making up 78,800 of the country’s population. 

'''''Religion'''''
*The country of Luxembourg does not have an official Religion even though the country is mainly broken down into 87% of people who practice the Catholic Religion. The remaining 13% of the population choose to practice other religions such as; Protestant, Jewish, and  Islamic.

'''''Ethnicity'''''
*The majority of the ethnic groups that live in Luxembourg, and are not native born, come from other European countries. The native born Luxembourger make up 63.1% of the population. The rest of the ethnic groups found in this country are 13.3% Portuguese, 4.5% French, 4.3% Italian, 2.3% Germans,7.3% other EU countries and, 5.2% of the rest of the population come from other countries. 

'''''Language'''''
*Luxembourgish is the national language of the country.  Languages like French and German, are also spoken in the country by its citizens.

'''''Economic development''''''
*''GDP per capital'' 
::In Luxembourg the GDP per capita is an estimated $79,600. 

*''GDP''-

*''Key industry''
::the key Industries in Luxembourg consists of  banking and financial services, iron and steel, information technology, telecommunications, cargo transportation, food processing, chemicals, metal products, engineering, tires, glass, aluminum, and tourism.

*''Exports''
::The Country of Luxembourg export chemicals, rubber products, glass, machinery and equipment. Netherlands, Italy, Belgium, Germany, France, and the UK are all export partners with Luxembourg. The country make about 14.18billion dollars in exports.

*''Imports''
::The Imports of Luxembourg consists of minerals, food products, quality consumer goods and metals. The Netherlands, China, France, Germany and Belgium are partners with Luxembourg. 


'''''Health and Education'''''
*The infant mortality rate is 4.56 deaths per 1,000 live births.
*The Life expectancies rate in the Country of Luxembourg is 79.33 years of the total population. Males in the country live to about an average age of 76.07, and the females live to about 82.81 years of age.
*The Literacy rate in Luxembourg is about 100%. In the country the people are exposed to a first- rate education system and the people of the country are required to stay in school until the age of 16, similar to the United States School System. 

https://www.cia.gov/library/publications/the-world-factbook/geos/lu.html
http://www.infoplease.com/ipa/A0107734.html?pageno=2

==Brief History==
In 1815 the Congress of Vienna made Luxembourg a Grand Duchy. Before this date Luxembourg was dominated by a lot of different European countries. It had been part of Charlimeagnes Empire, after being made a Grand Duchy the country was given to King William I in 1838. The country of Luxembourg became independent in 1835, but it was not until the year 1867 when Luxembourg was given the guarantee of perpetual neutrality, and recognized for their independence. In 1949 Luxembourg became a charter member of the North Atlantic Treaty Organization (NATO) and gave up their Neutrality.
During World War I, the country was used as an area where the German soldiers used to stay, even though the country of Luxembourg decided to remain neutral during the war. The same thing happened in World War II and the Germans once again took over the country of Luxembourg, during the war 5,259 of the citizens of Luxembourg lost their lives.

==Economic Development, Health, and Education==


==Governance==
The Luxembourg Government is set up like a Constitutional Monarchy. Luxembourg received its independence in 1839 from and adopted a constitution in 1868. Like the United States the Luxembourg government is broken down into three branches. The Executive Branch consists of the Grand Duke, who is the head of state, and is used only for ceremonial purposes. The Prime minister is also in this category and he is considered to be the head of state. The other branch of government is the legislative branch which is set up as a unicameral parliament; it has a chamber of deputies with council of state who serve as consultative body. The third branch of the Luxembourg Government is the Judicial Branch this is where the superior court is found. 


===Elections===


===Judicial Review===


==Courts and Criminal Law==

===Punishment===


===Legal Personnel===


===Law Enforcement===


The nation Police of Luxembourg is known as the “Police Grand-Ducale”. This name was adopted after January 2000. Before this date the Police system was made up into a dual system, the national police known as “corps de la police” and the Gendarmerie. The police work under the supervision of the Minister of Justice. 

A  general directorate, central units and regional divisions, are all part of the Police system in Luxembourg.
The General Directorate is divided into five different Branches and it has the overreaching authority.

The Secretary General- Watches over the five branches, buys the items for the different units and advices the new projects before the director General has time to make his decision.

Human services department is in charge of the recruitment, retirements, payroll, internal transfers and the training in foreign countries
The Operational department organizes big events. This department also takes care of the security intelligence, and deals with the function of the national emergency unit.

Budget and Equipment department handles the budget accounting, barracking and logistics covering the management of the fleet, arming, etc.
Information technology department assures the access and information of judicial documents, establishes criminal statistics, and runs the offices of international co-operation and the internal computer and telephone system.

Organization and Methodology department is where the creation and development of new procedures are sometimes created to replace the existing ones, it also has the responsibility of updating police documents.

The central Units is where the Criminal Investigation Department is Located. This department is organized based on the different types of crimes. The sub departments located in the Criminal Investigation Department include: Protection of Minors, Police Records, Crime Analysis General Crime, Organized Crime, Drugs, Financial and Economic Crimes. The Criminal Investigation Department also has the responsibility of preserving evidence and conserving crime scenes.

The Regional division is in charge of preventing crimes and misdemeanors.

===Crime Rates and Public Opinion===
Luxembourg is known to be a safe place. Violent crimes are not a common thing in this country. Theft on the other hand has slowly become a problem .Residential burglary happens in the country usually by people who spend a lot of time watching their victims every move in order to find the perfect opportunity to invade their home and steal their belongings. In public areas such as the hotel lobbies, train stations, bus stations and air ports, pick pocketing often occurs.  The most common non Violent Crime in Luxembourg is car theft. Lately there has been a new drug epidemic and the country is trying their best to try and cope with the situation. West African immigrants have been the ones eliciting the drugs to the youth who in return are committing petty crimes in order to support their habits. The country has been trying different things in order to deal with the problem. Outreach programs in the schools have proven to be effective so far.

==Rights==

===Family Law===

===Social Inequality===

===Human Rights===

===Works Cited===

&lt;references /&gt;

[[Category:Luxembourg]]</text>
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      <text bytes="433" sha1="rmu6es81slu0ogqumhv6ik4iy659lqa" xml:space="preserve">== [[Comparative_law_and_justice| Comparative Law and Justice]] Wiki Project ==
Countries with completed wiki pages that are located in Asia and the Middle East

[[../Afghanistan/]]

[[../China/]]

[[../India/]]

[[../Sri Lanka/]]

[[../Japan/]]

[[../Jordan/]]

[[../Oman/]]

[[../Philippines/]]

[[../Russia/]]

[[../Saudi Arabia/]]

[[../Thailand/]]

[[../United Arab Emirates/]]
[[Category:Comparative law and justice in Asia| ]]</text>
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&lt;td style=&quot;text-align:center;text-valign:middle;padding:4px 10px 10px 10px;background: white; color:var(--color-base-fixed,#202122);&quot;&gt;
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  &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot; align=&quot;center&quot;&gt;
  &lt;tr&gt;
  &lt;td&gt;&lt;hiero&gt;G5&lt;/hiero&gt;&lt;/td&gt;
  &lt;td height=&quot;48px&quot; width=&quot;2px&quot; style=&quot;background-color:black; color:white;&quot;&gt;&lt;/td&gt;
  &lt;td&gt;
    &lt;!--This table includes the horizontal lines and hieroglyphs of the Horus name--&gt;
    &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;&gt;
    &lt;tr&gt;
    &lt;td height=&quot;2px&quot; style=&quot;background-color:black; color:white;&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;&lt;tr&gt;
    &lt;td height=&quot;46&quot; style=&quot;padding: 0px 5px 0px 5px;&quot;&gt;{{{horus}}}&lt;/td&gt;
    &lt;/tr&gt;&lt;tr&gt;
    &lt;td height=&quot;2px&quot; style=&quot;background-color:black; color:white;&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;
    &lt;/table&gt;
  &lt;/td&gt;
  &lt;td align=&quot;center&quot; valign=&quot;middle&quot;&gt;[[Image:Srxtail2.svg]]&lt;/td&gt;
  &lt;/tr&gt;
  &lt;/table&gt;
&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:5px;background: white;&quot;&gt;''serekh'' or Horus name&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:4px 10px;background: white;&quot;&gt;
  &lt;!--This table surrounds the Nebty name--&gt;
  &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot; align=&quot;center&quot;&gt;
  &lt;tr&gt;
  &lt;td&gt;&lt;hiero&gt;G16&lt;/hiero&gt;&lt;/td&gt;
  &lt;td&gt;
    &lt;!--This table includes the horizontal lines and hieroglyphs of the Nebty name--&gt;
    &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;&gt;
    &lt;tr&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;td height=&quot;45&quot; style=&quot;padding: 0px 0px 0px 0px;&quot;&gt;{{{nebty}}}&lt;/td&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;
    &lt;/table&gt;
  &lt;/td&gt;
  &lt;/tr&gt;
  &lt;/table&gt;
&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:5px;background: white;&quot;&gt;''Nebty'' name&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:4px 10px 10px 10px;background: white;&quot;&gt;
  &lt;!--This table surrounds the Golden Horus name--&gt;
  &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot; align=&quot;center&quot;&gt;
  &lt;tr&gt;
  &lt;td&gt;&lt;hiero&gt;G8&lt;/hiero&gt;&lt;/td&gt;
  &lt;td&gt;
    &lt;!--This table includes the horizontal lines and hieroglyphs of the Golden Horus name--&gt;
    &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;&gt;
    &lt;tr&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;td height=&quot;45&quot; style=&quot;padding: 0px 0px 0px 0px;&quot;&gt;{{{goldenhorus}}}&lt;/td&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;
    &lt;/table&gt;
  &lt;/td&gt;
  &lt;/tr&gt;
  &lt;/table&gt;
&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:5px;background: white;&quot;&gt;''Golden Horus'' name&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:4px 10px;background: white;&quot;&gt;
  &lt;!--This table surrounds the praenomen--&gt;
  &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot; align=&quot;center&quot;&gt;
  &lt;tr&gt;
  &lt;td&gt;&lt;hiero&gt;M23-L2&lt;/hiero&gt;&lt;/td&gt;
  &lt;td&gt;[[Image:Hiero_Ca1.svg]]&lt;/td&gt;
  &lt;td&gt;
    &lt;!--This table includes the horizontal lines and hieroglyphs of the praenomen--&gt;
    &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;&gt;
    &lt;tr&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;&lt;tr&gt;
    &lt;td height=&quot;45&quot; style=&quot;padding: 0px 0px 0px 0px;&quot;&gt;{{{praenomen}}}&lt;/td&gt;
    &lt;/tr&gt;&lt;tr&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;/table&gt;
  &lt;/td&gt;
  &lt;td align=&quot;center&quot; valign=&quot;middle&quot;&gt;[[Image:Hiero_Ca2.svg]]&lt;/td&gt;
  &lt;/tr&gt;
  &lt;/table&gt;
&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:5px;background: white;&quot;&gt;''praenomen'' or throne name&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:4px 10px 10px 10px;background: white;&quot;&gt;
  &lt;!--This table surrounds the nomen--&gt;
  &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot; align=&quot;center&quot;&gt;
  &lt;tr&gt;
  &lt;td&gt;&lt;hiero&gt;G39-N5&lt;/hiero&gt;&lt;/td&gt;
  &lt;td&gt;[[Image:Hiero_Ca1.svg]]&lt;/td&gt;
  &lt;td&gt;
    &lt;!--This table includes the horizontal lines and hieroglyphs of the nomen--&gt;
    &lt;table border=&quot;0&quot; cellspacing=&quot;0&quot; cellpadding=&quot;0&quot;&gt;
    &lt;tr&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;&lt;tr&gt;
    &lt;td height=&quot;45&quot; style=&quot;padding: 0px 0px 0px 0px;&quot;&gt;{{{nomen}}}&lt;/td&gt;
    &lt;/tr&gt;&lt;tr&gt;
    &lt;td height=&quot;2px&quot; bgcolor=&quot;black&quot;&gt;&lt;/td&gt;
    &lt;/tr&gt;
    &lt;/table&gt;
  &lt;/td&gt;
  &lt;td align=&quot;center&quot; valign=&quot;middle&quot;&gt;[[Image:Hiero_Ca2.svg]]&lt;/td&gt;
  &lt;/tr&gt;
  &lt;/table&gt;
&lt;/td&gt;
&lt;/tr&gt;&lt;tr&gt;
&lt;td style=&quot;text-align:center;text-valign:middle;padding:5px;background: white;&quot;&gt;''nomen'' or birth name&lt;/td&gt;
&lt;/tr&gt;
&lt;/table&gt;
| era   = {{{era|default}}}
| style = {{{style|}}}
}}&lt;noinclude&gt;{{doc|content=

This table template produces a serekh and two cartouches for an Egyptian pharaoh's Horus name, praenomen and nomen. The parameters that can be set are align=left/right, era=sets the colour scheme, name=name in English of the pharaoh, horus=hieroglyphs for the Horus name with 'hiero' tags, praenomen=hieroglyphs of the pharaoh's praenomen with 'hiero' tags, nomen=hieroglyphs of the pharaoh's nomen with 'hiero' tags.

}}&lt;/noinclude&gt;</text>
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    <title>Moment distribution</title>
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      <text bytes="1250" sha1="ao4ni5op8x7vppmtama80jm0qh4ojo8" xml:space="preserve">Moment distribution method was developed by Hardy Cross to solve the problems of Indeterminate Structures. In this method all the joints are considered locked (restrained) in the beginning and the fixed end moments are determined at each joint. The stiffness coefficients are calculated which in turn determine the distribution factors for different members connected at a joint. 
: The next step is to release the locked joint and allow it to rotate (if not fixed in reality) and distribute that fixed-end-moment to the members connected at that joint in the proportion of distribution factors; this step is known as balancing of joint. This process continues till all the joints are balanced. When we unlock and joint and apply the balancing moment at that joint half of the moment will be developed at the far end which is known as carry-over moment. These cycle of balancing and carry-over continue till a desired accuracy is achieved at all the joint. This can be easily understood with the help of solved example&lt;ref&gt; [http://civilengineer.webinfolist.com/str/prob81.htm Solved Example on Indeterminate Beam] &lt;/ref&gt;

==References==
{{Reflist}}

==External Links==
* [http://civilengineer.webinfolist.com/str/mdm.htm Moment distribution method ]</text>
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  <page>
    <title>Understanding Arithmetic Circuits</title>
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      </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.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260410.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] ||
|| [[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>
      <sha1>a2vwhk198p3aapmavf6pm9slre2ihky</sha1>
    </revision>
    <revision>
      <id>2804323</id>
      <parentid>2804321</parentid>
      <timestamp>2026-04-11T13:42:33Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2804323</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3135" sha1="en8dqyg5xqzmqlhehgv98rd3j1il42z" xml:space="preserve">
== 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.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260411.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260304.pdf|B]] ||
|| [[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>
      <sha1>en8dqyg5xqzmqlhehgv98rd3j1il42z</sha1>
    </revision>
  </page>
  <page>
    <title>Template:Mathematics resources</title>
    <ns>10</ns>
    <id>152158</id>
    <revision>
      <id>2804372</id>
      <parentid>2746003</parentid>
      <timestamp>2026-04-11T19:19:27Z</timestamp>
      <contributor>
        <username>~2026-21801-82</username>
        <id>3064658</id>
      </contributor>
      <origin>2804372</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6572" sha1="1ryguea833hh6lmqklrdmlk6hjqmy8w" xml:space="preserve">{{Navbox
| name      = Mathematics resources
| title     = Mathematics resources
| bodyclass = hlist
| titlestyle = background:#F3E5AB; color:#000000;
| basestyle = background:#FAF0BE;
| evenstyle = background:#FFDEAD;
| oddstyle  = background:#D1BEA8;
| state     = {{{state&lt;includeonly&gt;|autocollapse&lt;/includeonly&gt;}}}
| groupstyle = background:#FAF0BE; color:#000000;
| belowstyle = background:#FAF0BE; color:#000000;

| group1 = Activities
| list1  = 
* [[Astrognosy/Laboratory|Astrognosy]]
* [[Alternative ways to become an observer|Becoming an observer]]
* [[Binary Stars and Extrasolar Planets|Binary stars and extrasolar planets]]
* [[Activity:Cassiopeia and Ursa Major|Cassiopeia and Ursa Major]]
* [[Astronomy/Craters/Laboratory|Cratering laboratory]]
* [[Distance to the Moon]]
* [[Electric orbits]]
* [[Electron beam heating/Laboratory|Electron beam heating]]
* [[International Year of Astronomy]]
* [[Lunar Boom Town]]
* [[Magnetic field reversals/Laboratory|Magnetic field reversal]]
* [[Observational astronomy]]
* [[Polar reversals]]
* [[Stars/Vega/Spectrum|Spectrum of Vega]]
* [[Standard candles/Laboratory|Standard candles]]
* [[Vertical precession]]
* [[Stars/X-ray classification/Laboratory|X-ray classification of a star]]
* [[X-ray trigonometric parallax/Laboratory|X-ray trigonometric parallax]]

| group2 = Articles
| list2  = 
* [[Radio Interferometer Telescope]]
* [[Changes in the properties of matter (mass spectrometer and spectral analysis of stars)|Spectral analysis of stars]]

| group3 = Categories
| list3  = 
* [[:Category:Algebra|Algebra]]
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* [[:Category:Secondary Math Lessons|Secondary Math Lessons]]
* [[:Category:Systems theory|Systems theory]]
* [[:Category:Trigonometry|Trigonometry]]

| group4 = Courses
| list4  = 
* [[Calculus I]]
* [[Calculus II]]
* [[Design and Analysis of Algorithms]]
* [[Foundations of mathematical concepts]]
* [[Geometry]]
* [[Information geometry]]
* [[Introduction to calculus]]
* [[Introduction to finite elements]]
* [[Introduction to Real Analysis]]
* [[Introduction to Statistical Analysis]]
* [[Introduction to Strategic Studies]]
* [[Introductory Algebra]]
* [[Mathematical Methods in Physics]]
* [[Our Playground: The Real Numbers and Their Development]]
* [[Ordinary differential equations]]
* [[The Real and Complex Number System]]
* [[Vector calculus]]

| group5 = Glossaries
| list5  = 
* [[Laboratory on Mathematics and Mathematics Education/Glossary]]

| group6 = Lectures
| list6  = 
* [[Abstract concept generator]]
* [[Actuarial Mathematics]]
* [[Actuarial science]]
* [[Applied analysis]]
* [[Astrophysics]]
* [[Calculus]]
* [[Christoffel symbols]]
* [[Discrete mathematics]]
* [[LaTeX]]
* [[Astronomy/Mathematics|Mathematical astronomy]]
* [[Mathematical induction]]
* [[Modelling]]
* [[Optimisation]]
* [[Probability]]
* [[Skewness]]
* [[Statistics]]
* [[Taylor's series]]
* [[Topology]]
* [[T-test]]
* [[Variable]]
* [[X-ray trigonometric parallax]]
* [[Z-test]]

| group7 = Lessons
| list7  = 
* [[Haskell/Lesson one]]
* [[Introduction to calculus - lesson 1|Introduction to Algebra; Introduction to Calculus./lesson01]]
* [[Ideas in Geometry/Analytic Geometry|Lesson Five: Analytic Geometry]]
* [[Factorising quadratics]]

| group8 = Lists
| list8  = 
* [[Materials Science and Engineering/List of Topics]]
* [[Physics Formulae|Tables of Physics Formulae]]

| group9 = Portals
| list9  = 
* [[Portal:Discrete Mathematics for Computer Science|Discrete Mathematics for Computer Science]]
* [[:Portal:Mathematics|Mathematics]]

| group10 = Problem sets
| list10  = 
* [[Angular momentum and energy]]
* [[Column densities]]
* [[Differential equations/Assignment 1]]
* [[Energy phantoms]]
* [[Furlongs per fortnight]]
* [[Lenses and focal length]]
* [[Nonlinear finite elements/Homework 1]]
* [[Planck's equation]]
* [[Radiation dosage]]
* [[Ideas in Geometry/Instructive examples/Section 1.2 problem|Section 1.2 problem]]
* [[Spectrographs]]
* [[Star jumping]]
* [[Synchrotron radiation/Problem set|Synchrotron radiation]]
* [[Telescopes and cameras]]
* [[Unknown coordinate systems]]
* [[Unusual units]]
* [[Vectors and coordinates]]

| group11 = Projects
| list11  = 
* [[Astronomy Project]]

| group12 = Quizzes
| list12  = 
* [[Abstract concept generator/Quiz]]
* [[Astrophysics/Quiz]]
* [[Calculus/Quiz]]
* [[Complex Analysis/Sample Midterm Exam 1]]
* [[Control groups/Quiz|Control group/Quiz]]
* [[Empirical astronomy/Quiz]]
* [[Astronomy/Mathematics/Quiz|Mathematical astronomy/Quiz]]
* [[X-ray trigonometric parallax/Quiz]]
* [https://www.derivativecalculus.com Derivative Calculus] - Interactive symbolic solver providing step-by-step differentiation rules for complex functions.

| group13 = Schools
| list13  = 
* [[:School:Biomathematics|Biomathematics]]
* [[:School:Mathematics|Mathematics]]

}}&lt;noinclude&gt;
[[Category:Mathematics templates]]
&lt;/noinclude&gt;</text>
      <sha1>1ryguea833hh6lmqklrdmlk6hjqmy8w</sha1>
    </revision>
    <revision>
      <id>2804373</id>
      <parentid>2804372</parentid>
      <timestamp>2026-04-11T19:24:33Z</timestamp>
      <contributor>
        <username>Saroj</username>
        <id>2896480</id>
      </contributor>
      <minor />
      <comment>Reverted edits by [[Special:Contribs/~2026-21801-82|~2026-21801-82]] ([[User talk:~2026-21801-82|talk]]) to last version by Koavf: unnecessary links or spam</comment>
      <origin>2746003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6415" sha1="72lvezaiivcn56atfza5pyy3l8y1dpm" xml:space="preserve">{{Navbox
| name      = Mathematics resources
| title     = Mathematics resources
| bodyclass = hlist
| titlestyle = background:#F3E5AB; color:#000000;
| basestyle = background:#FAF0BE;
| evenstyle = background:#FFDEAD;
| oddstyle  = background:#D1BEA8;
| state     = {{{state&lt;includeonly&gt;|autocollapse&lt;/includeonly&gt;}}}
| groupstyle = background:#FAF0BE; color:#000000;
| belowstyle = background:#FAF0BE; color:#000000;

| group1 = Activities
| list1  = 
* [[Astrognosy/Laboratory|Astrognosy]]
* [[Alternative ways to become an observer|Becoming an observer]]
* [[Binary Stars and Extrasolar Planets|Binary stars and extrasolar planets]]
* [[Activity:Cassiopeia and Ursa Major|Cassiopeia and Ursa Major]]
* [[Astronomy/Craters/Laboratory|Cratering laboratory]]
* [[Distance to the Moon]]
* [[Electric orbits]]
* [[Electron beam heating/Laboratory|Electron beam heating]]
* [[International Year of Astronomy]]
* [[Lunar Boom Town]]
* [[Magnetic field reversals/Laboratory|Magnetic field reversal]]
* [[Observational astronomy]]
* [[Polar reversals]]
* [[Stars/Vega/Spectrum|Spectrum of Vega]]
* [[Standard candles/Laboratory|Standard candles]]
* [[Vertical precession]]
* [[Stars/X-ray classification/Laboratory|X-ray classification of a star]]
* [[X-ray trigonometric parallax/Laboratory|X-ray trigonometric parallax]]

| group2 = Articles
| list2  = 
* [[Radio Interferometer Telescope]]
* [[Changes in the properties of matter (mass spectrometer and spectral analysis of stars)|Spectral analysis of stars]]

| group3 = Categories
| list3  = 
* [[:Category:Algebra|Algebra]]
* [[:Category:Algorithms|Algorithms]]
* [[:Category:Analysis|Analysis]]
* [[:Category:Applied mathematics|Applied mathematics]]
* [[:Category:Arithmetic‎|Arithmetic‎]]
* [[:Category:Basic mathematics|Basic mathematics]]
* [[:Category:Calculus|Calculus]]
* [[:Category:College algebra|College algebra]]
* [[:Category:Complex analysis‎|Complex analysis‎]]
* [[:Category:Complex numbers|Complex numbers]]
* [[:Category:Discrete mathematics‎|Discrete mathematics‎]]
* [[:Category:Elementary mathematics|Elementary mathematics]]
* [[:Category:Equations‎|Equations‎]]
* [[:Category:Finite element analysis|Finite element analysis]]
* [[:Category:Floating point|Floating point]]
* [[:Category:Functional analysis|Functional analysis]]
* [[:Category:Geometry|Geometry]]
* [[:Category:Laboratory on Mathematics and Mathematics Education|Laboratory on Mathematics and Mathematics Education]]
* [[:Category:Mathematical analysis|Mathematical analysis]]
* [[:Category:Mathematical physics|Mathematical physics]]
* [[:Category:Mathematical proofs|Mathematical proofs]]
* [[:Category:Mathematical theorems|Mathematical theorems]]
* [[:Category:Mathematics Media|Mathematics Media]]
* [[:Category:Numerical analysis|Numerical analysis]]
* [[:Category:Numerical methods|Numerical methods]]
* [[:Category:Olympiads|Olympiads]]
* [[:Category:Pre-Calculus‎|Pre-Calculus‎]]
* [[:Category:Probability|Probability]]
* [[:Category:Proofs|Proofs]]
* [[:Category:Pure Mathematics|Pure Mathematics]]
* [[:Category:Real numbers|Real numbers]]
* [[:Category:Representation theory|Representation theory]]
* [[:Category:School of Mathematics|School of Mathematics]]
* [[:Category:Secondary Math Courses|Secondary Math Courses]]
* [[:Category:Secondary Math Lessons|Secondary Math Lessons]]
* [[:Category:Systems theory|Systems theory]]
* [[:Category:Trigonometry|Trigonometry]]

| group4 = Courses
| list4  = 
* [[Calculus I]]
* [[Calculus II]]
* [[Design and Analysis of Algorithms]]
* [[Foundations of mathematical concepts]]
* [[Geometry]]
* [[Information geometry]]
* [[Introduction to calculus]]
* [[Introduction to finite elements]]
* [[Introduction to Real Analysis]]
* [[Introduction to Statistical Analysis]]
* [[Introduction to Strategic Studies]]
* [[Introductory Algebra]]
* [[Mathematical Methods in Physics]]
* [[Our Playground: The Real Numbers and Their Development]]
* [[Ordinary differential equations]]
* [[The Real and Complex Number System]]
* [[Vector calculus]]

| group5 = Glossaries
| list5  = 
* [[Laboratory on Mathematics and Mathematics Education/Glossary]]

| group6 = Lectures
| list6  = 
* [[Abstract concept generator]]
* [[Actuarial Mathematics]]
* [[Actuarial science]]
* [[Applied analysis]]
* [[Astrophysics]]
* [[Calculus]]
* [[Christoffel symbols]]
* [[Discrete mathematics]]
* [[LaTeX]]
* [[Astronomy/Mathematics|Mathematical astronomy]]
* [[Mathematical induction]]
* [[Modelling]]
* [[Optimisation]]
* [[Probability]]
* [[Skewness]]
* [[Statistics]]
* [[Taylor's series]]
* [[Topology]]
* [[T-test]]
* [[Variable]]
* [[X-ray trigonometric parallax]]
* [[Z-test]]

| group7 = Lessons
| list7  = 
* [[Haskell/Lesson one]]
* [[Introduction to calculus - lesson 1|Introduction to Algebra; Introduction to Calculus./lesson01]]
* [[Ideas in Geometry/Analytic Geometry|Lesson Five: Analytic Geometry]]
* [[Factorising quadratics]]

| group8 = Lists
| list8  = 
* [[Materials Science and Engineering/List of Topics]]
* [[Physics Formulae|Tables of Physics Formulae]]

| group9 = Portals
| list9  = 
* [[Portal:Discrete Mathematics for Computer Science|Discrete Mathematics for Computer Science]]
* [[:Portal:Mathematics|Mathematics]]

| group10 = Problem sets
| list10  = 
* [[Angular momentum and energy]]
* [[Column densities]]
* [[Differential equations/Assignment 1]]
* [[Energy phantoms]]
* [[Furlongs per fortnight]]
* [[Lenses and focal length]]
* [[Nonlinear finite elements/Homework 1]]
* [[Planck's equation]]
* [[Radiation dosage]]
* [[Ideas in Geometry/Instructive examples/Section 1.2 problem|Section 1.2 problem]]
* [[Spectrographs]]
* [[Star jumping]]
* [[Synchrotron radiation/Problem set|Synchrotron radiation]]
* [[Telescopes and cameras]]
* [[Unknown coordinate systems]]
* [[Unusual units]]
* [[Vectors and coordinates]]

| group11 = Projects
| list11  = 
* [[Astronomy Project]]

| group12 = Quizzes
| list12  = 
* [[Abstract concept generator/Quiz]]
* [[Astrophysics/Quiz]]
* [[Calculus/Quiz]]
* [[Complex Analysis/Sample Midterm Exam 1]]
* [[Control groups/Quiz|Control group/Quiz]]
* [[Empirical astronomy/Quiz]]
* [[Astronomy/Mathematics/Quiz|Mathematical astronomy/Quiz]]
* [[X-ray trigonometric parallax/Quiz]]

| group13 = Schools
| list13  = 
* [[:School:Biomathematics|Biomathematics]]
* [[:School:Mathematics|Mathematics]]

}}&lt;noinclude&gt;
[[Category:Mathematics templates]]
&lt;/noinclude&gt;</text>
      <sha1>72lvezaiivcn56atfza5pyy3l8y1dpm</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2804330</id>
      <parentid>2803940</parentid>
      <timestamp>2026-04-11T14:00:14Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2804330</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="ebqemsy6wntehyp7fevh7k3fr0ksv6h" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260408.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
      <sha1>ebqemsy6wntehyp7fevh7k3fr0ksv6h</sha1>
    </revision>
    <revision>
      <id>2804332</id>
      <parentid>2804330</parentid>
      <timestamp>2026-04-11T14:01:11Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2804332</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="jmtlqeuigxh697sijr4pas0s8n8250w" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260409.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
      <sha1>jmtlqeuigxh697sijr4pas0s8n8250w</sha1>
    </revision>
    <revision>
      <id>2804334</id>
      <parentid>2804332</parentid>
      <timestamp>2026-04-11T14:02:01Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2804334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="jmuckezczud1rd6gt3x0welav93wifp" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260410.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
      <sha1>jmuckezczud1rd6gt3x0welav93wifp</sha1>
    </revision>
    <revision>
      <id>2804336</id>
      <parentid>2804334</parentid>
      <timestamp>2026-04-11T14:05:34Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2804336</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="rym7s4e8esdcwfwl3kgvk6tg4tkxxn7" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260411.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Complex analysis]]</text>
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Welcome learners to Visayan 1, your aid in learning the Visayan dialect. This course is designed for English speakers. Knowing Visayan will help you succeed and aid you when visiting a Visayan-speaking region or learning another similar language, such as Spanish. 
{{center|1=&lt;big&gt;See also: [[Portal:Cebuano]]&lt;/big&gt;}}
Throughout this course, you will find fact sections that contain facts about Visayan-speaking countries and their cultures.  Our [[/forum|forum]] is a place where you can receive feedback and ask questions about Visayan.

[[Image:Misamis Oriental General Comprehensive High School.JPG|250px|right|thumb|Misamis Oriental General Comprehensive High School]]
[[Image:Visayan_languages_map.png|250px|right|thumb|Visayan speaking regions of the Philippines.]]
{{center|1=&lt;big&gt;See also: [[Southeast Asian Languages/Philippine Languages/Cebuano]]&lt;/big&gt;}}</text>
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&lt;span style=&quot;color: purple; font-size: large; line-height: .75em&quot;&gt;'''Understanding and improving our motivational and emotional lives&lt;br&gt;using psychological science (2016)'''&lt;/span&gt;]]&lt;/div&gt;
{{Motivation and emotion/Book/Quality}}
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| '''47th President'''
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[[../George W. Bush/]]

2001 – 2009
|[[File:President Barack Obama.jpg|100px]]

[[../Barack Obama/]]

2009 – 2017
|[[File:Donald Trump official portrait.jpg|100px]]

[[../Donald Trump/]]

2017 – 2021

|[[File:Joe Biden presidential portrait.jpg|100px|Joe Biden]]

[[../Joe Biden/]]

2021– 2025
|[[File:Official Presidential Portrait of President Donald J. Trump (2025).jpg|100px]]

[[../Donald Trump/]]

2025 – present
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| 45th President
| 46th President 
| '''47th President'''
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[[../George W. Bush/]]

2001 – 2009
|[[File:President Barack Obama.jpg|100px]]

[[../Barack Obama/]]

2009 – 2017
|[[File:Donald Trump official portrait.jpg|100px]]

[[../Donald Trump/]]

2017 – 2021

|[[File:Joe Biden presidential portrait.jpg|100px|Joe Biden]]

[[../Joe Biden/]]

2021– 2025
|[[File:Official Presidential Portrait of President Donald J. Trump (2025).jpg|100px]]

[[../Donald Trump/]]

2025 – present
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[[Category:Donald Trump|Trump, Donald]]</text>
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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]]) 20:47, 16 September 2021 (UTC)&lt;/div&gt;
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== [[Federal Writers' Project – Life Histories/2023/Fall/Section18/The Stembler Family (Minnie Stembler)]] ==

Hello, a page that you tagged for speedy deletion seems to be part of a school project that just started in this semester (as implied in the page directory). We can wait for a while for further development. If there are no changes in the next few months, then we will take care of the page. Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:17, 6 October 2023 (UTC)</text>
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== [[Federal Writers' Project – Life Histories/2023/Fall/Section18/The Stembler Family (Minnie Stembler)]] ==

Hello, a page that you tagged for speedy deletion seems to be part of a school project that just started in this semester (as implied in the page directory). We can wait for a while for further development. If there are no changes in the next few months, then we will take care of the page. Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:17, 6 October 2023 (UTC)</text>
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== [[Federal Writers' Project – Life Histories/2023/Fall/Section18/The Stembler Family (Minnie Stembler)]] ==

Hello, a page that you tagged for speedy deletion seems to be part of a school project that just started in this semester (as implied in the page directory). We can wait for a while for further development. If there are no changes in the next few months, then we will take care of the page. Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:17, 6 October 2023 (UTC)</text>
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      <comment>Reverted edits by [[Special:Contributions/Eridanus suplex|Eridanus suplex]] ([[User_talk:Eridanus suplex|talk]]) to last version by [[User:Quinlan83|Quinlan83]] using [[Wikiversity:Rollback|rollback]]</comment>
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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]]) 20:47, 16 September 2021 (UTC)&lt;/div&gt;
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== [[Federal Writers' Project – Life Histories/2023/Fall/Section18/The Stembler Family (Minnie Stembler)]] ==

Hello, a page that you tagged for speedy deletion seems to be part of a school project that just started in this semester (as implied in the page directory). We can wait for a while for further development. If there are no changes in the next few months, then we will take care of the page. Thank you for your attention. [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 13:17, 6 October 2023 (UTC)</text>
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      <comment>Mfield moved page [[User:MaRayneS/common.js]] to [[User:Marisatri/common.js]]: Automatically moved page while renaming the user &quot;[[Special:CentralAuth/MaRayneS|MaRayneS]]&quot; to &quot;[[Special:CentralAuth/Marisatri|Marisatri]]&quot;</comment>
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      <contributor>
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      <comment>Mfield moved page [[User talk:MaRayneS]] to [[User talk:Marisatri]]: Automatically moved page while renaming the user &quot;[[Special:CentralAuth/MaRayneS|MaRayneS]]&quot; to &quot;[[Special:CentralAuth/Marisatri|Marisatri]]&quot;</comment>
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&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] MaRayneS!''' 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.

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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]]) 03:18, 29 January 2022 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
      <sha1>hv7r5n33domtgtjjlhq6qoxjc2z25f2</sha1>
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  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2804326</id>
      <parentid>2803938</parentid>
      <timestamp>2026-04-11T13:56:12Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2804326</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="jy9mxm05ne13jm73q1ur5x1gihl2j5z" 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.20260410.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>jy9mxm05ne13jm73q1ur5x1gihl2j5z</sha1>
    </revision>
    <revision>
      <id>2804328</id>
      <parentid>2804326</parentid>
      <timestamp>2026-04-11T13:57:09Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2804328</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="pitz4rm9wzgs1cxnlbmrz3muns95phc" 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.20260411.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>
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    <title>User:Dc.samizdat/Real Euclidean four-dimensional space R⁴</title>
    <ns>2</ns>
    <id>289273</id>
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      <id>2804351</id>
      <parentid>2804192</parentid>
      <timestamp>2026-04-11T16:23:19Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The Kepler problem is framed in Euclidean 4-space */</comment>
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      <text bytes="122345" sha1="ax03zq4ohzxarvamyjnokgo4dpg3sti" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie in concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not in concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies in one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves on its own distinct geodesic spiral.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>ax03zq4ohzxarvamyjnokgo4dpg3sti</sha1>
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    <revision>
      <id>2804352</id>
      <parentid>2804351</parentid>
      <timestamp>2026-04-11T16:37:10Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122345" sha1="opl3voqyk4r9logit5vwx7rncjt1tu8" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves on its own distinct geodesic spiral.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>2804354</id>
      <parentid>2804352</parentid>
      <timestamp>2026-04-11T16:51:56Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122354" sha1="h05pf4j3bvbpr1vhy6q15o6hcxhdcp9" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every such 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves on its own distinct geodesic spiral.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>h05pf4j3bvbpr1vhy6q15o6hcxhdcp9</sha1>
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    <revision>
      <id>2804355</id>
      <parentid>2804354</parentid>
      <timestamp>2026-04-11T16:54:10Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122349" sha1="hhi1j83bpxnwoypf37vqniuat0xr2nh" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves on its own distinct geodesic spiral.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>hhi1j83bpxnwoypf37vqniuat0xr2nh</sha1>
    </revision>
    <revision>
      <id>2804356</id>
      <parentid>2804355</parentid>
      <timestamp>2026-04-11T17:05:32Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122484" sha1="rq6bfk5sm33ymighduk8nbwjmx19b1t" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of these two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>rq6bfk5sm33ymighduk8nbwjmx19b1t</sha1>
    </revision>
    <revision>
      <id>2804357</id>
      <parentid>2804356</parentid>
      <timestamp>2026-04-11T17:06:48Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122482" sha1="jhgrigws0wrkwq3a9o53huhnniu2za6" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a moving 3-dimensional manifold. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>jhgrigws0wrkwq3a9o53huhnniu2za6</sha1>
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    <revision>
      <id>2804358</id>
      <parentid>2804357</parentid>
      <timestamp>2026-04-11T17:11:32Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122509" sha1="jxcrf90mh75b59dcot8nrydoh75qpr3" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a 3-dimensional manifold moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 3-manifold, not necessarily a 3-sphere), then the Lorentz transformations 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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>jxcrf90mh75b59dcot8nrydoh75qpr3</sha1>
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    <revision>
      <id>2804359</id>
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      <timestamp>2026-04-11T17:27:55Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="122529" sha1="87qwgchj5baugqmd0x3h3qocjiw3iun" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 4 dimensions, even though we are physically confined to a 3-dimensional manifold moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>87qwgchj5baugqmd0x3h3qocjiw3iun</sha1>
    </revision>
    <revision>
      <id>2804360</id>
      <parentid>2804359</parentid>
      <timestamp>2026-04-11T17:31:30Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* A theory of the Euclidean cosmos */</comment>
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      <text bytes="122532" sha1="q59dqfxv355im1ljz3ututcrhhtljzy" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. Their direction of movement through space may be different for different observers (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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', in the same inertial reference frame. Your proper 4-space is rotated with respect to another observer's proper 4-space, precisely as your vectors (directions of motion) are rotated in Euclidean 4-space with respect to each other.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>q59dqfxv355im1ljz3ututcrhhtljzy</sha1>
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    <revision>
      <id>2804361</id>
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      <timestamp>2026-04-11T17:48:54Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="122755" sha1="jmx0zl1qxmh70ruv3jgnoirsjtzpwoy" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 is no metric distortion (no Lorentz transformations) between your coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>jmx0zl1qxmh70ruv3jgnoirsjtzpwoy</sha1>
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    <revision>
      <id>2804362</id>
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      <timestamp>2026-04-11T17:50:05Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Special relativity describes Euclidean 4-space */</comment>
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      <text bytes="122757" sha1="1kwp9k4pppi3fl8ot62686634cvhi0n" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

While this description may be understood as simple geometric pictures, some of the pictures may not be easy for us to visualize, since we have no physical experience in 4-dimensional space. &lt;small&gt;&lt;math&gt;\mathrm{R}, \mathrm{T}, \mathrm{Q}&lt;/math&gt;&lt;/small&gt; are just what they are in three-dimensional space, but &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt; is something new and unprecedented in our physical experience, because double rotations do not occur until you have four or more dimensions of space to rotate in.

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="122760" sha1="quqhxmjznp3elmbxvvlcj2q0kowaezw" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two or more double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>quqhxmjznp3elmbxvvlcj2q0kowaezw</sha1>
    </revision>
    <revision>
      <id>2804364</id>
      <parentid>2804363</parentid>
      <timestamp>2026-04-11T18:07:59Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Light propagates through 4-space at twice its apparent velocity c */</comment>
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      <text bytes="122752" sha1="2clj4587u329m7zc92dcjkjy7dgt9jm" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube.{{Efn|The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>2clj4587u329m7zc92dcjkjy7dgt9jm</sha1>
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    <revision>
      <id>2804365</id>
      <parentid>2804364</parentid>
      <timestamp>2026-04-11T18:10:22Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Light propagates through 4-space at twice its apparent velocity ''c'' */</comment>
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      <text bytes="122745" sha1="2rg8fiz0hhx13pkao3ny9j56iaadl2o" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; over the edge length (or radius) of the unit 4-hypercube. The 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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>2rg8fiz0hhx13pkao3ny9j56iaadl2o</sha1>
    </revision>
    <revision>
      <id>2804366</id>
      <parentid>2804365</parentid>
      <timestamp>2026-04-11T18:12:31Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Light propagates through 4-space at twice its apparent velocity c */</comment>
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      <text bytes="122761" sha1="4gn5txbajhvbxkqyfu7pynk1m57vxye" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>4gn5txbajhvbxkqyfu7pynk1m57vxye</sha1>
    </revision>
    <revision>
      <id>2804367</id>
      <parentid>2804366</parentid>
      <timestamp>2026-04-11T18:13:40Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Light propagates through 4-space at twice its apparent velocity c */</comment>
      <origin>2804367</origin>
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      <text bytes="122761" sha1="3jl2izb9eacaglfd0yypfupd17qqad8" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit, in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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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      <text bytes="122881" sha1="26h9j4p2rulirdx7rbe67agumoqftcy" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases typical of ordinary observations which agree closely with the predictions of classical physics, the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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>26h9j4p2rulirdx7rbe67agumoqftcy</sha1>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="122888" sha1="abp1w7pj8mx5d6jk5m48q4hemnbixy9" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases typical of most ordinary observations which agree closely with the predictions of classical mechanics, the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

Moreover, the real physical correlates of abstract orthogonal planes and rotation angles are very familiar to us viscerally in our body-language of physical experience, and we are also endowed with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions including rotations without even thinking about angles and orthogonal planes. This physical endowment amounts to an inborn capacity for dimensional analogy, since all our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful instinctive  visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space...

...cite Jesper Goransson's very concise paper

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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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        <username>Dc.samizdat</username>
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      <text bytes="122932" sha1="tp5y03224lillwosm4hijhvp2ymvq14" xml:space="preserve">= Real Euclidean four-dimensional space R⁴ =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|June 2023 - April 2026}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four orthogonal spatial dimensions. Space itself has a fourth orthogonal 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 our planet are only 3-dimensional objects, but nonetheless we can see in 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 has four perpendicular dimensions, not just three; atoms are [[W:4-polytope|4-polytopes]]; the sun is a 4-ball that is round in four dimensions; everything of intermediate size between an atom and a star, including us and our planet, lies in a 3-dimensional manifold of ordinary space; and our entire 3-space manifold is translating through Euclidean 4-space at the speed of light, in a direction perpendicular to its three interior dimensions.

== A theory of the Euclidean cosmos ==
The physical universe is properly visualized as a [[w:Four-dimensional_space|Euclidean space of four orthogonal spatial dimensions]]. Space itself has a fourth orthogonal dimension, of which we are unaware in ordinary life. Atoms are [[w:4-polytope|4-polytopes]], small round 4-dimensional objects, and stars are 4-balls of atomic plasma, large round 4-dimensional objects. Objects intermediate in size between atoms and stars, including molecules, people, and planets, 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 through Euclidean 4-space at 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 spiral galaxy such as ours is 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 galaxy appear from our viewpoint to be distributed in a cloud of elliptical spirals  occupying a flattened ellipsoid region of 3-dimensional space, but they are not so confined: they are distributed within a spherical region of 4-dimensional space. The galaxy's actual shape is spherical, not a flattened ellipsoid, but it is rounder than round can be in our ordinary experience: it occupies a hyperspherical region of space. The concentric spirals of stars that we observe lie on concentric [[W:3-sphere|3-sphere]]s (4-dimensional spheres), not on concentric 2-ellipsoids (3-dimensional elliptical spirals). Our sun and solar system lies on one of those concentric 3-spheres.

...rotating illustration of the 4-ball galaxy showimg its spirals of star clouds on the surface of concentric 3-spheres...obtained by reverse sterographic projection from 3D images of the galaxy...

The galaxy as a whole, or more properly its orbital center point, 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 relative to the galaxy's linear trajectory, as they pursue their various orbits within the galaxy. The spherical 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 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 cylinder; their trajectories are screw-displacements. 

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 a barycentric axis. It is the tremendous inertial force of stars in motion at velocity &lt;math&gt;c&lt;/math&gt; that holds the cylinder of motion together. 

The observed universe as a whole appears to be a 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. For all observers, the conjectured origin point of the universe 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). The 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 separately, 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 except the sun itself lie within this thinest three-dimensional manifold. 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.   

The enclosing surface of a spherical region of 4-space is itself a finite, curved (non-Euclidean) 3-dimensional space called a [[w:3-sphere|3-sphere]]. We live within such a 3-space, in an infinitesimally curved 3-manifold surface embedded in Euclidean 4-space. That surface is the ordinary 3-dimensional space we experience, and it contains the earth, all the planets and the 3-dimensional space between them. Our solar system is only a small patch on the 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 particular 3-sphere is one of the galaxy's concentric 3-spheres of spiral star-clouds. The solar system occupies a tiny patch of this filmy 4-dimensional soap-bubble of galactic size, that is thicker-skinned than the diameter of an atom only in the interior of stars and supermassive objects.  

Our entire 3-sphere manifold, as a spherical shell within the moving galaxy, is translating through 4-space at velocity &lt;math&gt;c&lt;/math&gt; with the galaxy in a distinct direction that is orthogonal to the manifold's three orthogonal dimensions of interior space. At every material point in the manifold (at every atom), the galaxy's translation is following a geometric law of motion discovered by Coxeter that governs the propagation of rotating objects through space by screw translation. 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 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 that is such an evolving surface boundary 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. Every object which we find occurring naturally in the solar system other than the sun itself, even the largest of 3-dimensional objects a planet, is a three-dimensional smear of atoms no thicker than one atom in its fourth dimension, which is the direction of movement through 4-space at velocity &lt;math&gt;c&lt;/math&gt; of the solar system's 3-manifold container, which is one of the galaxy's concentric 3-sphere shells.

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 little understanding of such 4-dimensional lumps thicker than one atom, since they occur naturally in our vicinity only in the interior of the sun. In fact the sun is the only such lump occurring naturally in our solar system. We refer to 4-dimensional lumps of matter as plasma, and have little experimental knowledge of their geometry or internal structure. We know that such a lump as 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 nothing about its interior 4-ball.

Every 3-dimensional surface boundary of matter in the observed universe is moving and evolving in four dimensions at velocity &lt;math&gt;c&lt;/math&gt;. Its current location in 4-space corresponds to the present moment in the proper time of its inertial reference frame. Its 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 atoms 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, two orthogonal motions each at the constant universal rate of transformation &lt;math&gt;c&lt;/math&gt;. Every object moves relative to universal 4-coordinate space on its own distinct geodesic spiral, a screw translation trajectory that is the compound of its two orthogonal inertial motions.  

Objects without mass such as photons lie off such surface boundaries of matter from which they were emitted, and their motion is of a different nature. They are in motion at velocity &lt;math&gt;c&lt;/math&gt; in all four dimensions concurrently, so they move diagonally 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 moving at velocity &lt;math&gt;c&lt;/math&gt;. For example, we can look across the center of our mostly-empty 4-ball galaxy and see stars in the opposite sides of its concentric 3-sphere 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; in the 4-space direction corresponding to their proper time, which is perpendicular to all three dimensions of their proper space. 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 their group actions express the principle of relativity. Here is Coxeter's formulation of the motions of objects (congruent transformations) possible in an ''n''-dimensional Euclidean space, excerpted:{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}
&lt;blockquote&gt;Let &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; denote a rotation, &lt;small&gt;&lt;math&gt;\mathrm{R}&lt;/math&gt;&lt;/small&gt; a reflection, &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; a translation, and let &lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r\mathrm{T}&lt;/math&gt;&lt;/small&gt; denote a product of several such transformations, all commutative with one another. Then &lt;small&gt;&lt;math&gt;\mathrm{RT}&lt;/math&gt;&lt;/small&gt; is a glide-reflection (in two or three dimensions), &lt;small&gt;&lt;math&gt;\mathrm{QR}&lt;/math&gt;&lt;/small&gt; is a rotary-reflection, &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is a screw-displacement, and &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is a double rotation (in four dimensions).&lt;br&gt;
Every orthogonal transformation is expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r \le n)&lt;/math&gt;&lt;/small&gt;, the number of dimensions.&lt;br&gt;
Transformations involving a translation are expressible as:&lt;br&gt;
:&lt;small&gt;&lt;math&gt;\mathrm{Q}^q \mathrm{R}^r \mathrm{T}&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
where &lt;small&gt;&lt;math&gt;(2^q + r + 1 \le n)&lt;/math&gt;&lt;/small&gt;.&lt;br&gt;
For &lt;small&gt;&lt;math&gt;(n = 4)&lt;/math&gt;&lt;/small&gt; in particular, every displacement is either a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}^2&lt;/math&gt;&lt;/small&gt;, or a screw-displacement &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; [where the rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; is a simple rotation, but the &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; is chiral like a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;]. Every enantiomorphous transformation in 4-space (reversing chirality) is a &lt;small&gt;&lt;math&gt;\mathrm{QRT}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; or a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, because we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;. By the same principle, we can view any &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; as an isoclinic (equi-angled) &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) for ''one'' of the two rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;) up to uncertainty, and can always distinguish the direction of their own proper time translation (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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 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;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The distinct Coxeter symmetry groups have characteristic &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&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, we shall see that stationary atoms exhibit the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetries of the discrete isoclinic (equi-angled) double rotations (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&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 &quot;space&quot; 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 to the three space coordinates so spacetime is hyperbolic, not a flat Euclidean 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 first described by Einstein himself in 1921, as a thought experiment in which he carefully described his fourth orthogonal spatial dimension as merely a mathematical abstraction.

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 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 ''c'', in some 4-space direction. 

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 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 ''c'', as measured by all observers in any inertial reference frame. This is the Lorentz invariant, that allows everyone to observe a constant speed of light, regardless of their motion relative to the light source. 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 ''c''. In physics as it has been universally understood, observers are not supposed to be able to move at velocity ''c''. 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 ''c'' through the universe, which is real Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&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 ''c''. 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 proper space, which again, may 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 proper space axes only if we are at rest with respect to each other, actually moving in the same direction at velocity ''c'', 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 coordinate systems; you are both embedded in the same Euclidean 4-dimensional space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.{{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 ''c'' relative to universal 4-coordinate space, so the maximum relative velocity between two observers is 2''c'' 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 ''c'', it is the same measurement in different units. Special relativity measures all velocities in a 3-space of Minkowski spacetime. Euclidean relativity measures all velocities in Euclidean 4-space.}}

So in this novel alternate view of relativity, every mass in the universe must be perpetually in motion at velocity ''c'' in 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; ''c'', 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 ''c'', but in at least slightly different directions. In Einstein's relativity, the invariant ''c'' is the speed of light through 3-space. In Euclidean relativity, the invariant ''c'' is the speed of matter through 4-space! The speed of light through 3-space is also perceived as ''c'' by all observers, because they are each living in a moving 3-manifold that is moving through 4-space at velocity ''c''. 

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 predictions about how observers in different reference frames will perceive each other's motions in time and space, and we shall see that they also agree on the predictions of general relativity. They both describe the same geometric relations of space and time, but they describe that geometry as embedded in two very different universal host spaces: Minkowski spacetime versus Euclidean 4-space.

...cite Lewis Epstein's elegant explanation of the Lorentz Invariance as observers moving at constant velocity &lt;math&gt;c&lt;/math&gt; through space and proper time

...cite Yamashita{{Sfn|Yamashita|2023}} on the equivalence of special relativity and Euclidean 4-space relativity

...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 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 ''c'' (on diverging paths), the propagation speed of light through 4-space between them would have to be greater than ''c''. Euclidean relativity is a revolutionary theory indeed, in which ''c'' cannot possibly be the speed of light!

We conclude that, for a theory of Euclidean 4-space to be physically viable (that is, for it 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 linearly along the edge of a unit 4-hypercube. This is conceivable in 4-space (and in no other Euclidean space of any dimensionality) because the diagonal of the unit 4-hypercube is the natural number &lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;.

== An object's motion in space is the 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 small number of discrete self-reflections. Any action of a geometric object that transforms its position and orientation in space may be measured as a distinct group of self-reflections of the object in its own surfaces. Any motion of the object whatsoever may be precisely described as the object propagating itself through space by a discrete set of local self-reflections.

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 surfaces which are parallel to each other. A rotation also occurs when an object self-reflects twice, but in two distinct surfaces which touch (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, leaving itself right-side-out again.

Coxeter's laws of motion are a geometric counterpart to Newton's laws of motion in three dimensional Euclidean space. They are helpful because they can be understood as simple geometric pictures, by anyone baffled by algebraic formulas. But they are also a revolutionary advance beyond Newton's laws, because Coxeter formulated them in Euclidean spaces of any number of dimensions. For example, they give us simple geometric pictures of all the possible motions of objects in four dimensional Euclidean space:

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

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

...to readers who have not studied Coxeter (almost all readers including TAC), the blockquote above is &quot;just math&quot;, not visualizable geometry...but I could describe Coxeter's congruent transformations in 4-space here geometrically: I could say clearly what they mean in spatial terms, in language anyone can understand, because they don't require any math to be understood; the &quot;math&quot; here is really just simple pictures (reflections and rotations); even double rotations can be visualized by dimensional analogy, as compounds of simple rotations...since even most physicists are unacquainted with Coxeter geometry, it really is important that I do this here...

== Light propagates through 4-space at twice its apparent velocity ''c''==
Coxeter's geometric laws of motion apply to all objects with mass in 4-dimensional Euclidean space, but 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;small&gt;&lt;math&gt;\mathrm{R}^4&lt;/math&gt;&lt;/small&gt;, which may be termed a double translation &lt;small&gt;&lt;math&gt;\mathrm{T}^2&lt;/math&gt;&lt;/small&gt;, a pure translation via two pairs of parallel reflections, without any rotation component &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;.

Matter (atoms and all particles with mass) are perpetually rotating and translating through 4-space by &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;, a screw translation of a rotating object, which is relativistically equivalent to a stationary isoclinic &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;, an isoclinically rotating object such as an atom. A simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; or simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; is a double reflection &lt;small&gt;&lt;math&gt;\mathrm{R^2}&lt;/math&gt;&lt;/small&gt;, so a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt; or &lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt; is also an &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;, but not with the same group of reflection angles as a light signal &lt;small&gt;&lt;math&gt;\mathrm{R^4}&lt;/math&gt;&lt;/small&gt;. A translation &lt;small&gt;&lt;math&gt;\mathrm{T = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two parallel planes, and a rotation &lt;small&gt;&lt;math&gt;\mathrm{Q = R^2}&lt;/math&gt;&lt;/small&gt; is a double reflection in two intersecting planes, as in a &lt;small&gt;&lt;math&gt;\mathrm{QT = R^4}&lt;/math&gt;&lt;/small&gt; which is both at once. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is two double reflections in pairs of parallel planes at once, a reflection in four or more non-intersecting parallel planes; it is all translation and no rotation. In a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; all the motion goes to translation, so the translation goes twice as far as the simple translation &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. A double translation &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&gt; is the opposite of a double rotation &lt;small&gt;&lt;math&gt;\mathrm{Q^2 = R^4}&lt;/math&gt;&lt;/small&gt;, which is stationary but rotates twice as fast as the simple rotation &lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt; in a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;. 

The product of the two translations in a &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; is a diagonal 4-space translation over the long diameter of the unit 4-hypercube, exactly twice the distance of a simple &lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&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-distance translation, but without any rotation component, so it cannot carry any mass with it. A &lt;small&gt;&lt;math&gt;\mathrm{T^2}&lt;/math&gt;&lt;/small&gt; cannot reposition a 4-polytope the way a &lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&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.

...lensing of double translations &lt;small&gt;&lt;math&gt;\mathrm{T^2 = R^4}&lt;/math&gt;&lt;/small&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...

== The Kepler problem is framed in Euclidean 4-space ==

The [[W:Kepler problem|Kepler problem]] is named for [[W:Johannes Kepler|Johannes Kepler]], arguably the greatest geometer since the ancients up to [[w:Ludwig Schläfli|Ludwig Schläfli]], who proposed [[W:Kepler's laws of planetary motion|Kepler's laws of planetary motion]] which solved the problem of the orbits of the planets, and investigated the types of forces that would result in orbits obeying those laws. Those forces were later identified by [[W:Isaac Newton|Isaac Newton]] in his[[W:Philosophiæ Naturalis Principia Mathematica| Principia]], where he proves what today might be called the &quot;inverse Kepler problem&quot;: the orbit characteristics require the force to depend on the inverse square of the distance.&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; 

The inverse square law behind the Kepler problem is the [[W:Central force|central force]] law which governs not only [[W:Newtonian gravity|Newtonian gravity]] and celestial orbits, but also the motion of two charged particles in [[W:Coulomb’s law|Coulomb’s law]] of [[W:Electrostatics|electrostatics]]; it applies to attractive or repulsive forces. Problems in which two bodies interact by a central force that varies as the [[W:Inverse square law|inverse square]] of the distance between them are called Kepler problems. Thus the [[W:Hydrogen atom|hydrogen atom]] is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law, another inverse-square central force. 

Using classical mechanics, the solution to a Kepler problem can be expressed as a [[W:Kepler orbit|Kepler orbit]] using six kinematical variables or [[W:Orbital elements|orbital elements]]. The solution conserves an orbital element called the [[W:Laplace–Runge–Lenz vector|Laplace–Runge–Lenz (LRL) vector]], a [[W:Constant of motion|constant of motion]], meaning that it is the same no matter where it is calculated on the orbit. The LRL vector was essential in the first quantum mechanical derivation of the [[W:Atomic emission spectrum|spectrum]] of the hydrogen atom, but this approach has rarely been used since the development of the [[W:Schrödinger equation|Schrödinger equation]]. The conservation of the LRL vector corresponds to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry, by Nother's theorem. The LRL vector lies orthogonal to both the orbital plane and the angular momentum vector of the Kepler orbit; we observe that it lies in a fourth orthogonal dimension. Fock in 1935&lt;ref&gt;V. Fock, Zur Theorie des Wasserstoffatoms, Zeitschrift für Physik. 98 (3-4) (1935), 145–154.&lt;/ref&gt; and Moser in 1970&lt;ref&gt;J. Moser, Regularization of Kepler’s problem and the averaging method on a manifold, Commun. Pure Appl. 23 (1970), 609–636&lt;/ref&gt; observed that the Kepler problem is mathematically equivalent to non-affine geodesic motion (a particle moving freely) on the surface of a 3-sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher-dimensional symmetry results in two well-known properties of the Kepler problem: the momentum vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points. 

...

Relativity establishes that an orbit in space is viewed in a different way in each distinct inertial reference frame. Depending on the choice of reference frame, the same Kepler system may be seen to be performing any one of a sequence of relativistically equivalent rotations in 4-space, on a continuum from an isoclinic rotation (Q&lt;sup&gt;2&lt;/sup&gt;) in the orbit's proper reference frame, to a screw transfer (QT) with a simple rotation component (Q) and a translation component (T) at velocity &lt;math&gt;c&lt;/math&gt;, in the universal reference frame of 4-coordinate space wherein every object is seen to be translating at velocity &lt;math&gt;c&lt;/math&gt;. In reference frames between these two limit cases, the orbit is seen to be performing a double rotation (Q&lt;sup&gt;2&lt;/sup&gt;) at two unequal, completely orthogonal angular rates of rotation: an elliptical double rotation. These include the reference frames of most typical observers, who are moving slowly relative to the observed orbital system's reference frame (their relative motion is a small fraction of the speed of light). In these cases typical of most ordinary observations which agree closely with the predictions of classical mechanics, the non-isoclinic elliptical (Q&lt;sup&gt;2&lt;/sup&gt;) resembles a (QT), because one of its two completely orthogonal rotations (Q) has such a long period that it is almost indistinguishable from a straight translation (T).

All orbits in 4-space are isoclinic in their own reference frame. Orbiting objects in their own proper Kepler systems follow circular geodesic isoclines through 4-space. Orbits in 4-space are perfectly circular in their own reference frame, as Copernicus assumed the orbits of planets to be. It is the orbit's path through the 3-space of its elliptic hyperplane that is an ellipse, as Kepler found it to be.

...cite Jesper Goransson's very concise paper

The geodesic circle that an orbiting object follows through 4-space in the proper reference frame of its own Kepler system is not a simple great circle which turns in two orthogonal dimensions. It is a helical great circle that turns in four orthogonal dimensions at once.{{Efn|Geodesic orbits in 4-space are not simple 2-dimensional great circles; they are helical 4-dimensional great circles that curve in all four dimensions at once. Their circular trajectories are helixes which we call ''isoclines'', since they are the paths taken by points on a rigid object undergoing isoclinic rotation.}} Such circles lie outside our physical experience, since our local space has only three orthogonal dimensions. Nonetheless we can visualize them in imagination, because their helical, circular shape is perfectly well defined by the kinematical variables of the Kepler orbit. 

The real physical correlates of abstract orthogonal planes and rotation angles are already familiar to us viscerally in our body-language of physical experience, since we are endowed biologically with highly evolved visual signal processing engines. These enable us to see and understand spatial relations and motions, including rotations, without even thinking about angles and orthogonal planes. This physical endowment is an inborn capacity for dimensional analogy which our biologic evolution has provided. All our instinctive spatial reasoning is by dimensional analogy from flat 2-dimensional retinal images to 3-dimensional scenes, using our powerful inborn visualization capacities of reverse stereographic projection and pattern recognition. We humans are thus very well equipped with everything we need to see in four-dimensional space, except experience.

...

Recently Anco and Moghadam found that through Noether’s theorem in reverse, the LRL vector gives rise to a corresponding infinitesimal dynamical symmetry on the kinematical variables, which they show to be the semi-direct product of &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, in contrast to the &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; symmetry group generated by the LRL symmetries and the rotations.{{Sfn|Anco|Moghadam|2026|ps=; The physically relevant part of the LRL vector is its direction  ... since its magnitude is just a function of energy and angular momentum.}} This remarkable symmetry breaking is expressive of the ''dimensional relativity'' between ordinary 3-space &lt;small&gt;&lt;math&gt;\mathbb{R^3}&lt;/math&gt;&lt;/small&gt;, spherical space &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt; and Euclidean space &lt;small&gt;&lt;math&gt;\mathbb{R^4}&lt;/math&gt;&lt;/small&gt;.

Consider a hydrogen atom in a Kepler orbit: for example, a hydrogen atom moving freely in space in an orbit around the sun. It is a ''double'' Kepler problem: an electrostatic Kepler problem within itself, and a gravitational Kepler problem in its environment. 

The ''single'' electrostatic Kepler problem of a hydrogen atom moving freely in space beyond any gravitational influence is a problem in special relativity. In our Euclidean 4-space model, this atom viewed as stationary in its own proper reference frame exhibits an &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry corresponding to an isoclinic double rotation (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). The fourth dimension in this reference frame is the atom's proper time vector; it has constant velocity &lt;math&gt;c&lt;/math&gt; and constant direction. From the point of view of our universal 4-coordinate space (which cannot be the proper inertial reference frame of any physical observer, all of whom are moving relative to it at velocity ''c''), the entire Kepler system (the atom) is translating through 4-space via a screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) at constant velocity &lt;math&gt;c&lt;/math&gt;. From this viewpoint the atom has only a simple &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;), breaking its stationary &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; isoclinic rotation symmetry (&lt;small&gt;&lt;math&gt;\mathrm{Q^2}&lt;/math&gt;&lt;/small&gt;). Because each discrete part of the rotating atom moves along a helical trajectory through 4-space, the atom is in orbit around a barycentric axis (like a star in a galaxy), but only in a tiny orbit within its own radius, which is its inertial domain of rotation. The straight 4-dimensional cylinder it progresses along at velocity &lt;math&gt;c&lt;/math&gt; is very narrow: only the diameter of the rotating atom itself.

The gravitational Kepler problem of a hydrogen atom in a Kepler orbit around the sun is a problem in general relativity. In our 4-space model, this atom viewed in its own proper reference frame exhibits the same &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; rotation symmetry as it did in the electrostatic Kepler problem where the atom was translating linearly through space. The Kepler system in this case is not just the atom; it is the entire solar system. The LRL vector of this Kepler system is the proper time vector of the atom's inertial reference frame; once again it has constant velocity ''and constant direction''. Although the momentum vector moves in a perfect circle as the atom orbits the sun, the 4-space LRL vector does not move at all: it is a constant of motion, of linear motion (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) of the Kepler system (the entire solar system in this case) in a constant 4-space direction, the proper time direction of the system. The direction of the system's proper time vector would vary under some kinds of acceleration of the atom, but it is constant under this kind of orbital acceleration. It continues to point in the same direction, like a 4-space compass needle, as the atom winds its way along its spiral path around the axis of the sun's straight-line translation through 4-space at velocity &lt;math&gt;c&lt;/math&gt;. This compass needle always points in the direction the sun is moving, not the direction the atom is moving at any instant.

...Its Kepler orbit around the sun is its &lt;small&gt;&lt;math&gt;SO(3)&lt;/math&gt;&lt;/small&gt; rotation component (&lt;small&gt;&lt;math&gt;\mathrm{Q}&lt;/math&gt;&lt;/small&gt;).

Although the atom is moving on a geodesic circle in the second problem, by the [[equivalence principle]] the difference in the state of the atomic systems in these two problems cannot be observed by examining the atoms alone. Even from another inertial reference frame, where the atom in the second problem is seen to be translating through 4-space via a wide screw translation (&lt;small&gt;&lt;math&gt;\mathrm{QT}&lt;/math&gt;&lt;/small&gt;) around the sun's axis of motion, there is still no difference between the two problems which can be detected by examining only the atoms within their own proper reference frames (even over time), because the LRL vector (&lt;small&gt;&lt;math&gt;\mathrm{T}&lt;/math&gt;&lt;/small&gt;) is a constant of motion of the entire system in both cases.

...Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;) breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;)... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt;) ... Minkowski spacetime if the energy is positive (a hyperbolic orbit).

...

Finally we consider a third problem in which a hydrogen atom enters the solar system as a comet, loops around the sun and exits the solar system again. This atom...

...

As Hamilton found when he discovered the quaternions, we see that it is necessary to admit a fourth dimension to the system in order to properly model the problem: in Hamilton's case the general problem of ..., and in our case the Kepler problem. These are instances of the same problem in 4-dimensional Euclidean geometry, and indeed a solution to the Kepler problem in quaternions (the four Cartesian coordinates of Euclidean 4-space) is a solution to it in our model of the 4-coordinate Euclidean cosmos.

== Distribution of stars in our galaxy ==

The stars in our own galaxy appear to us to be a rotating spiral cluster in 3-dimensional space. By assuming that light from them reaches us on straight lines through space, by assuming that we can measure their distance from us by its red shift, and by assuming that they are distributed in three dimensions of space, we have plotted their locations in 3-space. If we abandon the last of those three assumptions, we can just as easily reinterpret that dataset to plot their distribution around us in 4-dimensional space, and see how they actually lie. 

When we perform this experiment on the data for the stars in our galaxy, do we indeed find that they are distributed non-uniformly in various concentric spirals, but the spirals lie on the surface of various 3-spheres, rather than in elliptical orbits as we saw them in 3-space? That would be an expected consequence of the special rotational symmetry group of 4-space &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt;, in which circular (isoclinic) orbits are the geodesics (shortest rotational paths) rather than elliptical (non-equi-angled double rotation) orbits.

...have to perform this experiment somehow, at least as a conclusive thought experiment, before I publish this paper...

== Rotations ==
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Great hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways in the completely orthogonal plane of rotation (like coins flipping) into each other's planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat great circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one of the parallel planes of rotation, so all the vertices move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a [[24-cell#Rotations|detailed description]] enabling the reader to properly visualize its counter-intuitive consequences runs to many pages and illustrations, with many accompanying pages of explanatory notes on surprising phenomena that arise in 4-dimensional space: [[24-cell#Great squares|completely orthogonal planes]], [[24-cell#Clifford parallel polytopes|Clifford parallelism]]{{Efn|name=Clifford parallels}} and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Isoclinic rotations|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a unique surprise. [[User:Dc.samizdat/Rotations#Sequence of regular 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120 and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (with one exception), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. None of these symmetries is to be found in 3-dimensional space, although their simpler 3-dimensional analogues are all present there. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It subsumes 3-dimensional space, with all of the symmetries we are accustomed to, and adds astonishing new surprises. These are hard for us to visualize, because the only way we can experience them is in our imagination; we have no body of sensory experience in 4-dimensional space to draw upon, other than our evolution in time.

For that reason (our difficulty in visualizing them), descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a single rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case, which is easiest for us to understand. Compound and [[W:Kinematics|kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rotation of a single rigid 24-cell. 

To begin with, when we examine the individual parts of a single rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, then in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertex positions just once, and no point-object colliding with (or even crossing the path of) any other at any time. This is an example of a discrete Hopf fibration. But it is still an example of a rigid object in a discrete isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing one half of the characteristic rotation of the 24-cell. 

We can also imagine ''combining'' distinct isoclinic rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rigid rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible in the same 3-sphere shell without collisions? In adjacent concentric shells without asymmetric imbalance? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore such questions of [[W:kinematics|kinematics]], and where dynamic stabilities arise, of [[wikipedia:kinetics (physics)|kinetics]].

In four dimensions, we discover that space has more room in it than we have experienced, which permits previously unimagined motions. Even 3-space is more commodious than we thought; when it is curved and lies embedded in a higher-dimensional space, it permits previously impossible symmetric packings. Sadoc studied double-twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space as the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack on the 3-sphere perfectly without exhibiting any torsion, although their packing in ordinary flat 3-space is imperfect, &quot;frustrated&quot; by their twisted geometry.
&lt;blockquote&gt;The frustration, which arises when the molecular orientation is transported along the two [spiral] AB paths of figure 1 [double twist helix], is imposed by the very topological nature of the Euclidean space R&lt;sup&gt;3&lt;/sup&gt;. It would not occur if the molecules were embedded in the non-Euclidean space of the [[W:3-sphere|3-sphere]] S&lt;sup&gt;3&lt;/sup&gt;, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers, along which the molecules can be aligned without any conflict between compactness and [[W:torsion of a curve|torsion]].... The fibres of this [[W:Hopf fibration|Hopf fibration]] are great circles of S&lt;sup&gt;3&lt;/sup&gt;, the whole family of which is also called the [[W:Clifford parallel|Clifford parallel]]s.{{Efn|name=Clifford parallels}} Two of these fibers are C&lt;sub&gt;∞&lt;/sub&gt; symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.{{Efn|name=helical geodesic}} These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S&lt;sup&gt;3&lt;/sup&gt;.{{Efn|name=Petrie polygon of a honeycomb}} They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint.{{Sfn|Sadoc &amp; Charvolin|2009|loc=§1.2 The curved space approach|ps=; studies the helical orientation of molecules in crystal structures and their imperfect packings (&quot;frustrations&quot;) in 3-dimensional space.}}&lt;/blockquote&gt;

Of course we do not find molecules condensing to close-pack the 3-sphere in our experience, and Sadoc does not say that we do. We find 3-spheres in the atomic realm (if atoms are 4-polytopes), and in the cosmic realm (as the surface boundaries of stars, and the concentric surfaces of galaxies). But in between, in the realm of ordinary experience which includes the molecular realm, ourselves and all the objects we can materially handle or observe up close including the planets, we are confined together by gravity as inertia within a curved 3-dimensional space that is no more than one atom thick in the fourth spatial dimension. That is why in the molecular realm we find only objects that occupy 3-spaces which, though infinitesimally curved in the fourth dimension, are tiny patches on whole 3-spheres of galactic size. So Sadoc's exercise is a thought experiment, like Einstein's gedankenexperiments about railroad embankments and trains moving at nearly the speed of light. It is no less illuminating, despite the symmetry it reveals not having a realization as an actual 3-sphere of actual molecules. And might not something very like it have an actual realization in the atomic realm? 

We know that atoms have their own complex internal structure, which we are unable to model geometrically in ordinary 3-dimensional space. Suppose such a model is impossible because an atom is actually a 4-polytope occupying a tiny spherical region of 4-dimensional space, and so we only find its constituent particles in close-packed helical orbits on the 3-sphere, in the manner of Sadoc's imaginary twisted molecules, but as real 4-dimensional helices of atomic scale. We would expect to find the atomic orbit of a fundamental particle in some discrete Hopf fibration characteristic of a symmetry group, that is, on the maximally symmetric isoclines of a discrete isoclinic rotation characteristic of some regular 4-polytope and the particle.

== A theory of the Euclidean atom ==
&lt;blockquote&gt;Because quantum physics could be tested without being understood, it allowed humans to see how the universe worked without knowing why.&lt;ref&gt;Sebastian Junger, In My Time of Dying&lt;/ref&gt;&lt;/blockquote&gt;
...

== Light and Mass are Reflection and Rotation ==

The phenomena of light and mass are expressions of reflection symmetries and rotation symmetries, respectively.

...

Atoms are 4-polytopes, elementary objects with SO(4) rotational symmetry.

Light is ....

Motion in space is the propagation of the elementary objects of light and matter in Coxeter congruent transformations by kaleidoscopic self-reflections, like the motion of self-reproducing cellular automata in [[Conway's Game of Life|Conway's game of life]].

...

=== Atoms are 4-polytopes ===

...

== Relativity in real space of four or more orthogonal dimensions ==

Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is Galilean relativity in a general space of four or more orthogonal dimensions, e.g. in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, and any orthogonal 4-manifold.

Light is a consequence of symmetry group reflections at quantum scale. Gravity and the other fundamental forces are consequences of rotations, which are consequences of quantum reflections. Both kinds of motion are group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer may properly see themself as stationary and the universe as an ''n''-sphere with themself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and can be measured by the observer as the speed of light.

=== Special relativity is Galilean relativity in a Euclidean space of four orthogonal dimensions ===

...TAC suggests this section is needed sooner, i.e. in the preceding Special Relativity section, as it explains how Euclidean relativity reduces special relativity to 4D perspective geometry...it's misplaced (too late) here...

Perspective effects known as the Lorentz transformations occur because each observer's proper 3-dimensional space is a moving curved manifold embedded in flat 4-dimensional Euclidean space. The curvature of their 3-space complicates sightline calculations for observers; they sometimes require Lorentz transformations to produce the actual 4-space 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, except when an observer wants to calculate a projection, that is, the shadow of how things will appear to them from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} Space really has four orthogonal dimensions, and space and time behave there just as they do in a classical vector space, only 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 three dimensions 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), and all observers and all scenes being observed are in motion (at constant velocity) in their respective proper time directions, we observe perspective foreshortenings in time as well as in three spatial dimensions. In special relativity these perspective effects are reciprocal, precisely because they are only apparent, not actual, changes in size and duration. (In general relativity, discussed below, the actual rate of physical processes varies from place to place, and those differences are neither reciprocal nor illusory.)

None of these Lorentz effects are beyond geometric explanation or paradoxical.  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 than Euclidean 3-space, analogous to the way 3-space is much more interesting and 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, successive Euclidean spaces are dimensionally 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, is an exact expression of intrinsic symmetries: that is Nother's great discovery.

=== General relativity is Galilean relativity in a general space of four orthogonal dimensions ===

...

== Dimensional relativity ==
Coxeter's kinetic law of &lt;math&gt;n&lt;/math&gt;-dimensional congruent Euclidean transformations may be called ''dimensional relativity'', since it captures the theories of special and general relativity entire, 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 accessable by reason of direct analogy, as [[w:Ludwig Schläfli|Ludwig Schläfli]] subsequently demonstrated.

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 spherical and Euclidean &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 degree of dimensional analogy of which they are capable, 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 relativity may perceive the universe from any inertial reference frame, not only from their own proper perspective. We see that every observer may properly view themself as stationary and the universe as an ''n''-sphere with themself at the center observing it, perceptually equidistant from all points on its 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. It 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 and spherical spaces of any number of dimensions. 

It should be known as Thoreau's principle of ''spherical relativity'', since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{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.]}}

== 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 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 ''c'', 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, or 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 we dispersed those stars into an infinite Euclidean 3-space, and relinquished our privileged position at the center. Then Einstein showed us 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 sense related to time, at least, it must be 4-dimensional. In this work we suggest 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, the universe viewed as a 4-sphere might, or might not, be expanding, and 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 now observe them, any more than they lie in the three dimensional hyperplane of local space in which we find everything residing in our solar system.

When we look up at the objects that surround us, we have no way of discerning how many dimensions beyond three the space we are looking into has. We know their distance from us only by virtue of how long it takes their light to reach us. We can measure their distribution around us in 4-space, but that is simply how we choose to measure them, not a finding of how they are actually 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 ''n''-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 much closer observation will reveal evidence of whether everything we see lies in the same 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 displacements), nearby objects must be translating radially at the same 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 displacement 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 each galaxy's own distinct 3-sphere 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 starship Earth, to boldly go where no one has gone before. We made the jump to lightspeed 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 the geometry of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below).{{Efn|[[W:William Rowan Hamilton|Hamilton]]'s algebra '''H''' of [[W:Quaternions|quaternions]] contains the notion of a [[W:Three-dimensional sphere|three-dimensional sphere]] embedded in a four-dimensional space, but Hamilton did not conceive of the quaternions as the Cartesian 4-coordinates of a Euclidean 4-space, and did not describe our ordinary 3-space 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.&lt;ref&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; 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;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of 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 spacetime of four dimensions. 

Anco and Maghadam found that &lt;small&gt;&lt;math&gt;SO(4)&lt;/math&gt;&lt;/small&gt; breaks to ... &lt;small&gt;&lt;math&gt;S^3&lt;/math&gt;&lt;/small&gt;... if the energy in the Kepler orbit is negative (an elliptical orbit), and to ... &lt;small&gt;&lt;math&gt;H^3&lt;/math&gt;&lt;/small&gt; ... Minkowski spacetime if the energy is positive (a hyperbolic orbit). Because the planets orbit on ellipses in our 3-space, Euclidean 4-space is the actual geometry of our physical universe, and Minkowski spacetime is an abstraction; the reciprocal of Einstein's disclaimer is the truer model. Of course spacetime remains a true and useful abstraction, although it must relinquish its privileged position of centrality as our exclusive conception of our place in space.

...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 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 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 four-dimensional 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 motions and objects. 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. A boundary is an 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 and Coxeter'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 in imagination and then 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. This project is forever beginning anew. Coxeter showed us that Schläfli's Euclidean space is an expression of intrinsic symmetries, as Noether showed us all of physics is. Kappraff and Adamson discovered that even the sequences of humble regular polygons have fractal complexity, and Conway found that symmetry itself is chaotic, always reachable but forever beyond our complete grasp. We are on a Wilderness Project, 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. 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|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{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}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{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}}
* {{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:}}
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==Early history of the twin paradox==

{{Lorentzbox|Text={{center|Date of article creation: 9 November 2023; Last major revision: 2 March 2026}}}}

a) When was the [[:w:twin paradox]] applied to life forms and human beings?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt;&lt;ref group=S name=during /&gt; report that {{slink||Einstein 1911-HU}} discussed the aging of living organisms, and that {{slink||Langevin 1911-HU}} and {{slink||Wiechert 1911-HU}} explicitly discussed the aging of human beings. 

:*More details in sections {{slink||Human beings in 1911|Twins from 1911 to 1920}}, including newspaper articles from 1911 written by {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}} that clearly show that Einstein was the first to explicitly discuss the aging of human beings as well.

b) Who was the first to formulate the principle of maximal proper time along straight worldlines, upon which differential aging in the standard twin paradox is based?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=during /&gt; mention Langevin (1911), Laue (1911). 

:*More details in section {{slink||Maximal proper time}} with the contributions of Langevin (1911), Wiechert (1911), Study (1911), Laue (1911-13).

c) Who was the first to formulate [[w:Triangle inequality#Reversal in Minkowski space|inverse triangle inequality]] in Minkowski space, which represents the simplest version of the twin paradox?

:*See details in section {{slink||Triangle inequality}} with the contributions of Robb (1914-20), Eddington (1922), Rogers (1922).

d) Who was the first to show that any influence of proper acceleration on clocks can be neglected in the computation of the twin paradox from the viewpoint of the stay-at-home twin?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt; mention Einstein (1911), Laue (1913). 

:*More details in section {{slink||Negligibility of proper acceleration}} with the contributions of Einstein (1911), Wiechert (1911), Laue (1913), Lorentz (1913).

e) Who was the first to introduce the three clock/brother example that completely removes acceleration from the clock/twin paradox?

:*Historical accounts&lt;ref group=S name=debs /&gt;&lt;ref group=S name=alizzi /&gt; date it back to Lange (1927) and Lord Halsbury (1957).

:*More details in section {{slink||Relay (three brothers) experiment}} with the contributions of Grünbaum (1911) and Wiechert (1920-22).

f) Who was the first to use acceleration as an asymmetry indicator?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref name=cuvaj group=S /&gt;&lt;ref group=S name=pes /&gt; mention Langevin (1911), Einstein (1918). 

:*More details in section {{slink||Acceleration as asymmetry indicator}} with the contributions of Langevin (1911), Sommerfeld (1913), Lorentz (1913), Einstein (1914-20).

g) Who was the first to use different frame distribution as asymmetry indicator as an asymmetry indicator?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=pes /&gt; mention Laue (1911-13). 

:*More details in section {{slink||Frame distribution as asymmetry indicator}} with the contributions of Laue (1911-13), Bloch (1918).

h) Who was the first to describe the perspective of the traveler?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=beng /&gt; mention Langevin (1911), Lorentz (1914), Einstein (1918). 

:*More details in section {{slink||Perspective of the traveler}} with the contributions of Langevin (1911), Lorentz (1913-14), Einstein (1918), Thirring (1921).

i) Who was the first to describe a round-trip experiment in curved spacetime?

:*See section {{slink||Curved spacetime}} with the contribution of Becquerel (1922).

j) Who was the first to denote the round-trip experiment as paradoxical?

:*Historical accounts&lt;ref group=S name=miller /&gt;&lt;ref group=S name=during /&gt; point to Laue (1911).

:*See section {{slink||Paradoxical?}} for details.

k) Who was the first to misunderstand the twin paradox?

:*See section {{slink||Misunderstandings}} with the contributions of Berg (1910), Wiechert (1911), Campbell (1911/12), Gruner (1912).

==Human beings in 1911==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

![[w:Albert Einstein|Einstein]] 

|-

|{{anchor|Einstein 1905}}In 1905&lt;ref name=einstein05 /&gt; he showed that a clock moving on a round-trip away from A and back along a polygonal or curved path, is retarded with respect to a clock stationary at A by approximately &lt;math&gt;\tfrac{1}{2}t(v/V)^{2}&lt;/math&gt; at reunion. For example, a clock on the equator is retarded with respect to a clock on the pole. He described this consequence as being &quot;peculiar&quot; (German: eigentümlich).

{{anchor|Einstein 1911-HU}}In a lecture given on January 1911&lt;ref name=einstein11a /&gt; (published in November), he extended this &quot;funny&quot; (German: drollig) experiment to living organisms:

{| 

! width=55% | Einstein wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Wenn wir z. B. einen lebenden Organismus in eine Schachtel hineinbrächten und ihn dieselbe Hin- und Herbewegung ausführen lassen wie vorher die Uhr, so könnte man es erreichen, dass dieser Organismus nach einem beliebig langen Fluge beliebig wenig geändert wieder an seinen ursprünglichen Ort zurückkehrt, während ganz entsprechend beschaffene Organismen, welche an den ursprünglichen Orten ruhend geblieben sind, bereits längst neuen Generationen Platz gemacht haben. Für den bewegten Organismus war die lange Zeit der Reise nur ein Augenblick, falls die Bewegung annähernd mit Lichtgeschwindigkeit erfolgte!

| style=&quot;padding: 0px 20px 0px 20px;&quot; |For example, if we put a living organism in a box and make it undergo the same back and forth movement as the clock before, we could achieve that this organism returns to its original location with arbitrary little change after a flight of arbitrary length, whereas completely identical organisms that remained at rest in the original location have long since made room for new generations. To the moving organism, the long journey was only a moment if the movement happened close to the speed of light!

|}

{{Lorentzbox|Text=Two participants of that lecture, {{slink||Lämmel 1911-HU}} and {{slink||Müller 1911-HU}}, report that Einstein also talked about the aging of ''human beings''.}}

|-

!{{anchor|Lämmel 1911-HU}}[[w:Rudolf Lämmel|Lämmel]]

|-

|He attended Einstein's 1911 lecture and gave a popular report about it in the Swiss newspaper &quot;[[w:Neue Zürcher Zeitung|Neue Zürcher Zeitung]]&quot; published on 28 April 1911,&lt;ref name=lammel /&gt; including additional details. Regarding the round-trip clock experiment he wrote:

{| 

! width=50% | Lämmel wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Bewegt sich eine Uhr mit Lichtgeschwindigkeit längs einer Geraden, auf der gerichtete Uhren stehen, so scheint die bewegte Uhr, beurteilt vom Standpunkt der ruhenden aus, im oben stizzierten Sinn, stillzustehen. Kehrt die Uhr, nach einem Ruck, mit Lichtgeschwindigkeit wieder zurück zur Zentral-Uhr, so ist, nach Einstein, für den Beobachter bei der Zentral-Uhr die Sache so, als ob ein mit der bewegten Uhr mitgeführter Beobachter (samt dessen Uhr) nicht gealtert hätte. Hinge also des letzteren Alter von den Angaben des ruhenden Beobachters ab, so könnte der von einer großen Reise ins Weltall zurückkehrende Beobachter bei der Zentral-Uhr spätere Generationen antreffen – er selber hätte nicht gealtert. Welche Bedeutung diese ''ad absurdum'' geführte Gedankenspielerei etwa hat, läßt sich heute nicht absehen – vielleicht, ja wahrscheinlich ist sie ohne jeden Einfluß auf die tatsächlichen Verhältnisse. Aber man sieht dabei immerhin, daß die Physik imstande ist, die kühnsten Träume der Phantasie noch – zu überbieten.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Let a clock be moving at speed of light along a line on which regulated clocks are standing, then the moving clock's hand appears to be standing still (in the sense described above) as judged from the standpoint of the resting one. If the clock, after one jolt, comes back with light speed to the central clock, then according to Einstein the matter presents itself to the observer at the central clock, as if the observer comoving with the clock (together with his clock itself) hasn't been grown older. Thus if the age of the latter would depend on the indications of the resting observer, the observer returning from a great journey into space could meet later generations at the central-clock – he himself hasn't been grown older. The importance of this play of thought led ''ad absurdum'' cannot be seen today – maybe, or even probably, it is without any influence on the actual situations. Though at least one can see that physics is able to – surpass – even the boldest dreams and fantasies.

|}

Lämmel in December 1920 (published 1921)&lt;ref name=lammel2 /&gt; again alluded to Einstein's lectures in Zürich (possibly the one from 1911, and maybe also later ones), describing a discussion between himself and Einstein. After Einstein concluded that the travelers who came back after their journey will probably meet their former contemporaries as old men while they themselves could have been away for only a few years, Lämmel objected that this conclusion is only drawn with respect to rods and clocks, but not with respect to living beings. Einstein responded though, that all processes in the blood, in the nerves etc. are eventually periodical oscillations, i.e. motions. Yet to any such motion the relativity principle applies, thus the conclusion regarding the unevenly rapid aging it permissive.

{{Lorentzbox|Text=While the official publication of Einstein's January lecture ({{slink||Einstein 1911-HU}}) mentions the aging of organisms, Lämmel recalls the reference to the aging of a human space traveler (&quot;observer returning from a great journey into space&quot;). This means that Einstein was the first to use human beings in the clock/twin paradox on January 16 which was first published by Lämmel on April 28, 1911. In comparison, {{slink||Langevin 1911-HU}} used space travelers in a lecture on April 10 with publication in July, and {{slink||Wiechert 1911-HU}} used space travelers in lectures held between March 25 and May 23 with publication in July/September. It seems very unlikely that before April 28, Lämmel became somehow aware of the content of Langevin's or Wiechert's lectures held a few weeks earlier, in order to use them in his description of Einstein's lecture.}}

|-

!{{anchor|Langevin 1911-HU}}[[w:Paul Langevin|Langevin]]

|-

|On 10 April 1911, published July 1911,&lt;ref name=langevin1 /&gt; he held a now famous lecture popularizing the clock/twin paradox which he derived from the proper time integral as described in {{slink||Langevin 1911-PT}}. He demonstrated that a moving radioactive sample of radium is less evolved and less aged and therefore more active at return then the ones that remained in the laboratory. He also used light signals and the Doppler effect to visualize the effect. The most famous part concerned his description of the aging of human space travelers:

{| 

! width=50% | Langevin wrote

! [[:s:Translation:The Evolution of Space and Time|English Wikisource translation]]

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Cette remarque fournit le moyen, à celui d’entre nous qui voudrait y consacrer deux années de sa vie, de savoir ce que sera la Terre dans deux cents ans, d’explorer l’avenir de la Terre en faisant dans la vie de celle-ci un saut en avant qui pour elle durera deux siècles et pour lui durera deux ans, mais ceci sans espoir de retour, sans possibilité de venir nous informer du résultat de son voyage puisque toute tentative du même genre ne pourrait que le transporter de plus en plus avant.

Il suffirait pour cela que notre voyageur consente à s’enfermer dans un projectile que la Terre lancerait avec une vitesse suffisamment voisine de celle de la lumière, quoique inférieure, ce qui est physiquement possible, en s’arrangeant pour qu’une rencontre, avec une étoile par exemple, se produise au bout d’une année de la vie du voyageur et le renvoie vers la Terre avec la même vitesse. Revenu à la Terre ayant vieilli de deux ans, il sortira de son arche et trouvera notre globe vieilli de deux cents ans si sa vitesse est restée dans l’intervalle inférieure d’un vingt-millième seulement à la vitesse de la lumière. Les faits expérimentaux les plus sûrement établis de la physique nous permettent d’affirmer qu’il en serait bien ainsi.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |This remark provides the means for any among us who wants to devote two years of his life, to find out what the Earth will be in two hundred years, and to explore the future of the Earth, by making in his life a jump ahead that will last two centuries for Earth and for him it will last two years, but without hope of return, without possibility of coming to inform us of the result of his voyage, since any attempt of the same kind could only transport him increasingly further.

For this it is sufficient that our traveler consents to be locked in a projectile that would be launched from Earth with a velocity sufficiently close to that of light but lower, which is physically possible, while arranging an encounter with, for example, a star that happens after one year of the traveler's life, and which sends him back to Earth with the same velocity. Returned to Earth he has aged two years, then he leaves his ark and finds our world two hundred years older, if his velocity remained in the range of only one twenty-thousandth less than the velocity of light. The most established experimental facts of physics allow us to assert that this would actually be so.

|}

{{Lorentzbox|Text=Reading his lecture in full, one finds the word &quot;paradoxical&quot; only in relation to the constancy of light speed, not on relation to the round-trip clock experiment.}}

|-

!{{anchor|Wiechert 1911-HU}}[[w:Emil Wiechert|Wiechert]]

|-

|In lectures on 25 March and 23 May 1911, submitted July and published September 1911,&lt;ref name=wiechert11 /&gt; he described the round-trip clock experiment with two equal clocks regulated to the same rate and brought to the same pointer position, or by introducing the same chemical process two times, or by introducing ''two life forms that began their life at the same time''. At the end of his paper he applied this to human travelers:

{|

! width=50% | Wiechert wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Nehmen wir aber wieder eine Relativgeschwindigkeit an, die bis auf 3 Proz. der Lichtgeschwindigkeit nahekommt, dann wird das Verhältnis der empfundenen Zeitlängen wie 4:1. Das Bild mag etwas weiter noch ausgemalt werden. Denken wir uns, daß ein Beobachter durch den Raum unseres Sternhimmels mit dieser Geschwindigkeit in einer Kreisbahn mit einem Radius von 16 Lichtjahren fährt, dann wird er nach unserer Zeitrechnung nach je 100 Jahren wieder an unserem Sonnensystem vorüberkommen. In seinem Gefährt wird dabei die Zentrifugalkraft so auf ihn einwirken, daß sie gemäß den Relativitätsgesetzen der Einwirkung der Schwerkraft auf uns Erdenbewohner gleich erscheint. Es sind also die wirkenden Kräfte nur so groß, daß der Phantasie die Möglichkeit geboten wird, den Reisenden als menschliches Wesen zu denken. Da hier dauernd &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; ist, fließt die Eigenzeit für den Reisenden viermal langsamer dahin, als für die Bewohner der Gestirne. Wenn er also nach 100 unserer Jahre wieder zu unserem Sonnensystem zurückkehrt, wird er sich selbst nur um 25 Jahre gealtert fühlen. Erreicht er nach der Entwicklung seines Körpers und nach seiner Zeitempfindung ein Alter von 75 Jahren, so entspricht dies doch einer dreimaligen Wiederkehr zu unserem Sonnensystem, also 300 unserer Erdenjahre.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Yet if we again assume a relative velocity approximating the speed of light by 3 percent, then the ratio of the experienced duration of time becomes 4:1. This image can be further extended. Let's imagine that an observer travels with that velocity on a circular path at a radius of 16 light years through the space of our galaxy, then according to our time calculation he passes by our solar system every 100 years. In his vehicle the centrifugal force will act on him in such a way, that in accordance with the relativity laws it will appear to be equal to the force of gravity acting upon the inhabitants of Earth. Thus the acting forces are only thus big, in order to give our fantasy the possibility to imagine the traveler as a human being. Since we have &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; throughout, proper time flows four times slower for the traveler than for the inhabitants of the stars. Thus when he comes back to our solar system after 100 of our years, he will feel to have aged only by about 25 years. If he reaches an age of 75 years according to the development of his body and his own time experience, then this corresponds to a threefold return to our solar system, i.e. 300 of our Earth years.

|}

{{Lorentzbox|Text=a) Wiechert (1915)&lt;ref name=wiechert15 /&gt; later provided a short historical survey of the clock/twin paradox. He referred to the fact that already {{slink||Einstein 1905}} considered the case of two clocks (&quot;Einstein's clock experiment&quot;), and even though [[w:Hermann Minkowski|Minkowski]] himself didn't consider the case, his proper time formula provides the result in a straight forward manner. The latter was done by himself in lectures on 25 March and 23 May 1911, as well as by Langevin published in July 1911. Wiechert pointed out that he himself and Langevin used &quot;humorist&quot; examples in order to clarify the situation: While Wiechert argued that one has to make a journey in order to stay young, Langevin argued that one has to romp about in a laboratory in order to stay young. Both of them used human beings, arguing that their physical and mental life should have been influenced in the same way as any other process in nature.

b) The dates given by Wiechert (1915) are not complete. The correct ones are:

*Langevin's lecture on 10 April 1911, published in July.

*Wiechert's lectures on 25 March and 23 May 1911, submitted on July 26, published in September.

*He was still unaware of Einstein's lecture from January 1911, published in November 1911.}}

|-

!{{anchor|Müller 1911-HU}}[[w:Fritz Müller-Partenkirchen|Müller]]

|-

|The freelance writer and law student Fritz Müller (who was later known as [[w:Fritz Müller-Partenkirchen|Müller-Partenkirchen]]) attended Einstein's lecture and wrote a popular report about it in the German newspaper &quot;[[w:Berliner Tageblatt|Berliner Tageblatt]]&quot; on 16th and 23rd October 1911,&lt;ref name=muller /&gt; in which he gave further details (compare with {{slink||Lämmel 1911-HU}}). Regarding the clock/twin paradox he wrote:

{| 

! width=50% | Müller wrote

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Zwei gleichgehende Uhren sollen je einen Beobachter haben und nebeneinander ruhen. Nun soll die eine mit ihrem Beobachter plötzlich mit Lichtgeschwindigkeit in den Weltenraum hinausreisen. Vorher haben die beiden vereinbart, sich alle Sekunden mit einem Lichtsignal die Zeit zu telegraphieren. [...] In unserem Grenzfall, wo die Reise mit Lichtgeschwindigkeit vor sich geht, müßte der ruhende Beobachter erklären, jene andere Uhr käme in der Zeit überhaupt nicht voran. Die Zeit stünde dort still. Tatsächlich kommen die Einsteinschen Gleichungen zu diesem Resultat. Für den mit der Uhr reisenden Beobachter, sagt Einstein, gelte dasselbe. Das heißt, im Urteil des Zurückbleibenden würde jener niemals alt. „Und wenn er auf einer gebrochenen Reiselinie wieder an seinen Ausgangspunkt zurückkehrte?&quot; fragt man den Vortragenden in der Diskussion. – „So bliebe er in unserem Urteil so jung wie bei der Ausreise,&quot; erwidert Einstein mit vollem Ernst, „selbst wenn wir Zurückgebliebenen inzwischen Männer mit weißen Bärten geworden sind – die Gleichungen liefern für jede Richtung der Bewegung, auch für eine gebrochene Bewegung, unerschütterlich die selben Resultate.&quot; – Wir sehen einander an. Das klingt märchenhaft. Märchenhaft? Gewiß, die alten Märchen vom Mönch von Heisterbach, vom Rip van Winkle, von Urashima Taro steigen auf. Merkwürdig, wie die Volksphantasie bei den Deutschen, bei den Amerikanern, bei den Japanern in der gleichen Richtung gearbeitet hat – alle drei Märchen erzählen ja von Leuten, deren Leben still steht, viele hundert Jahre lang, während die andern altern. So fanden sie bei ihrer Rückkehr ein anderes Land und eine andere Generation.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Two synchronous clocks at rest next to each other, shall each be accompanied by an observer. Now one of them, together with its observer, suddenly travels into space at the speed of light. Previously, both have arranged that every second they telegraph their time to each other using light signals. [...] In our limiting case where the journey happens at light speed, the resting observer would have to declare that the other clock would not proceed in time at all. Time would stand still at this place. Einstein's equations indeed produce this result. As to the observer traveling with the clock, says Einstein, the same is true. That means in the judgment of the remaining one, the other one would never become old. Then the lecturer [i.e. Einstein] was asked in the discussion: &quot;And if he comes back to his starting point on a curved travel path?&quot;, to which Einstein replied in full earnest: &quot;Then in our judgment he would remain as young as he was at departure, even if we remaining ones became men with white beards in the meantime, the equations unshakably give the same result in every direction of motion, also for curved motion&quot;. We look at each other. That sounds fabulous. Fabulous? Of course, the old fairy tales of [[w:Heisterbach Abbey|w:The monk of Heisterbach]] or [[w:Rip Van Winkle]] or [[w:Urashima Tarō]] come forward. Strange, how the folk fantasy of the Germans, the Americans, the Japanese worked in the same direction, all three fairy tales indeed tell about people whose life stands still, many hundred years long, while the other ones grow old. Thus they found another country and another generation when they returned.

|}

{{Lorentzbox|Text=Müller's account confirms {{slink||Lämmel 1911-HU}} that Einstein indeed mentioned human beings, but his description also suggests that Einstein was the first to use mutually sent light signals. However, as this was published in October, it cannot be excluded that Müller's description of light signals was influenced by {{slink||Langevin 1911-HU}}, published in July, in which light signals were used as well.}}

|}

==Twins from 1911 to 1920==

We now provide a list of authors who employed ''twins'', i.e. ''two'' life forms or humans that initially were of ''same age'' when the round-trip began:

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Date !! Description

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

|1911

|Two life forms that begin their life at the ''same time'' (German: &quot;Zwei Lebewesen [..] die ihr Leben gleichzeitig beginnen&quot;), of which the moving one returns retarded in its progression with respect to the stationary one.

|-

|[[w:Paul Gruner|Gruner]]&lt;ref name=gruner /&gt;

|1912

|Two persons of ''same age'' (French: &quot;deux personnes du même âge&quot;), of which the moving one returns less developed than stationary one.

|-

|[[w:Max von Laue|Laue]]&lt;ref name=laue3 /&gt;

|1913

|The moving life form returns younger than its ''former agemates'' (German: &quot;ehemaligen Altersgenossen&quot;).

|-

|[[w:Hermann Weyl|Weyl]]&lt;ref name=weyl /&gt;

|Easter 1918

|

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Von zwei Zwillingsbrüdern, die sich in einem Weltpunkt A trennen, bleibe der eine in der Heimat (d. h. ruhe dauernd in einem tauglichen Bezugsraum), der andere aber unternehme Reisen, bei denen er Geschwindigkeiten (relativ zur »Heimat«) entwickelt, die der Lichtgeschwindigkeit nahekommen; dann wird sich der Reisende, wenn er dereinst in die Heimat zurückkehrt, als merklich jünger herausstellen denn der Seßhafte.

|Suppose we have two twin-brothers who take leave from one another at a world-point A, and suppose one remains at home (that is, permanently at rest in an allowable reference-space), whilst the other sets out on voyages, during which he moves with velocities (relative to “home”) that approximate to that of light. When the wanderer returns home in later years he will appear appreciably younger than the one who stayed at home. 

|}

{{Lorentzbox|Text=Weyl was the first to ''explicitly use twins'' in relation to the round-trip experiment. The fourth edition (1920) of that book was translated from German into English and French in 1922.}}

|-

|[[w:Albert Einstein|Einstein]]&lt;ref name=einstein20 /&gt;

|1920/21

|{{Anchor|Einstein 1921-TW}}

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Trifft A wieder bei B ein, so kann es sich ereignen, daß der beharrende Zwilling inzwischen 60 Erdjahre alt geworden ist, während der zurückkehrende nur 15 Jahre zählt, oder sich gar noch im Säuglingsstadium befindet. [..] Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls -Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. 

|If A then returns to B, it may happen that the twin who stayed at home is now sixty years old, whereas the wanderer is only fifteen years of age, or is perhaps only an infant still. [..] In the case of these two twins, Einstein declared, we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures. 

|}

{{Lorentzbox|Text=This was based on an interview of Einstein by Moszkowski. While the expression &quot;clock paradox&quot; was used since 1911/12 (see section {{slink||Paradoxical?}}), this seems to be the first time that it was rebranded as &quot;twin paradox&quot;. The copyright mark indicates 1920, while the title page indicates 1921. The translation from German into English also appeared in 1921.}}

|}

==Maximal proper time==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Paul Langevin|Langevin]]

1911

|{{anchor|Langevin 1911-PT}}In April 1911 (published July),&lt;ref name=langevin1 /&gt; he described the round-trip experiment without formulas using two portions of matter present at two events happening at the same place. The ''integration of proper time'' along the entire wordlines shows that the portion of matter that starts a closed cycle by receding and finally coming back, will have a ''smaller proper time'' than the one that stayed behind.

In October 1911 (published 1912),&lt;ref name=langevin2 /&gt; Langevin again showed that the portion of matter that described a closed cycle will have a ''smaller proper time'' &lt;math&gt;R&lt;/math&gt; than the one that stayed in an inertial frame, which is defined by the equation:

:&lt;math&gt;\begin{matrix}V^{2}\left(t-t_{0}\right)^{2}=d^{2}-R\\
\left[d^{2}=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]
\end{matrix}&lt;/math&gt; 

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

Lectures March-May 1911

submitted July

published September

|{{anchor|Wiechert 1911-PT}}Let two equal processes be observed in two equal material systems colocated in two moments (1) and (2), and let there velocities have been changed in arbitrarily different ways in the meantime. It follows that the ratio of advancement of those processes is given by the two intervals &lt;math&gt;\Delta\tau &lt;/math&gt; of their respective ''proper times''. He concluded that any round-trip clock experiment can be easily comprehended from that theorem by computation. The corresponding integral is:

:&lt;math&gt;\Delta\tau=\int_{1}^{2}d\tau=\int_{1}^{2}dt\sqrt{1-\frac{\mathfrak{v}^{2}}{c^{2}}}&lt;/math&gt; 

|-

|[[w:Eduard Study|Study]]&lt;ref name=study /&gt;

June 1911

|Minkowski's concept of worldlines implies that the straight path between two points of the same worldline is the ''longest'' among all paths between those points, if the path length on a worldline is defined by the related proper time.

{{Lorentzbox|Text=Study's book was purely mathematical without mentioning clocks or the round-trip experiment, alluding to his result only in a footnote.}}

|-

|[[w:Max von Laue|Laue]]

1911-13

|{{anchor|Laue 1911/12-PT}}In December 1911 (published 1912),&lt;ref name=laue1 /&gt; Laue showed without formulas that the round-trip experiment is represented by a curved worldline, which at worldpoint A decomposes into a row of curves, after which all of them will be re-united at worldpoint B to a single line. Of all curves connecting the points A and B having time-like direction throughout, the straight connection has the ''longest proper time.''

{{anchor|Laue 1912/13-PT}}In December 1912 (published 1913) in the second edition of this relativity book,&lt;ref name=laue1 /&gt; Laue described the proper time integral between events 1 and 2 of a slowly accelerated clock covering a broken line and a stationary clock covering a straight worldline. Of all worldlines covering 1 and 2, the straight line has the ''longest proper time''. Therefore the traveling clock in the round-trip experiment is retarded at reunion, because its curved worldline corresponds to a shorter proper time. This result he presented in terms of the following inequality, of which the right-hand side refers to the straight curve of the stationary clock, while all others possible curves are represented on left-hand side:

:&lt;math&gt;\tfrac{1}{c}\int_{1}^{2}\sqrt{du^{2}-\left(dx^{2}+dy^{2}+dz^{2}\right)}&lt;\tfrac{1}{c}\int_{1}^{2}du&lt;/math&gt; 

{{Lorentzbox|Text={{anchor|Sommerfeld 1913-PT}}Similar treatments can be found in the textbooks of [[w:Arnold Sommerfeld|Sommerfeld]] (1913),&lt;ref name=sommerfeld /&gt; [[w:Hermann Weyl|Weyl]] (1918),&lt;ref name=weyl /&gt; [[w:Wolfgang Pauli|Pauli]] (1921),&lt;ref name=pauli /&gt; [[w:August Kopff|Kopff]] (1921),&lt;ref name=kopff /&gt; [[w:Jean Becquerel|Becquerel]] (1922).&lt;ref name=becqu1 /&gt;}}

|}

==Triangle inequality==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|valign=top|[[w:Alfred Robb|Robb]]

1914-1920

|{{anchor|Robb 1914-TR}}In 1914&lt;ref name=robb1 /&gt; he showed that there are three types of triangles formed by intervals in Minkowski space, depending on whether one deals with &quot;separation lines&quot; (spacelike intervals), &quot;optical lines&quot; (lightlike intervals), or &quot;inertia lines&quot; (timelike intervals representing the path of nonaccelerated particles defined by &lt;math&gt;{\scriptstyle \left(x_{1}-x_{0}\right)^{2}+\left(y_{1}-y_{0}\right)^{2}+\left(z_{1}-z_{0}\right)^{2}-c^{2}\left(t_{1}-t_{0}\right)^{2}&lt;0}&lt;/math&gt;). As to a triangle formed by inertia lines, he showed that the sum of a certain two sides is ''less'' than that of the third one. 

{{Lorentzbox|Text=So the triangle inequality derived from time-like intervals in Minkowski space is ''[[w:Triangle inequality#Reversal in Minkowski space|inverse]]'' to the inequality in Euclidean space. This inverse inequality directly represents the most simple variant of the twin paradox: the traveler follows two sides of the time-triangle, while the stay-at-home observer follows the third side indicating maximal proper time.}}

[[File:RobbTriangle.svg|right|150px]]
In 1920&lt;ref name=robb2 /&gt; Robb gave a numerical example of the triangle ABC with time-like intervals (&quot;inertia lines&quot;) defined by coordinates

:&lt;math&gt;\begin{matrix} &amp; x &amp; y &amp; z &amp; t\\
A\  &amp; 0 &amp; 0 &amp; 0 &amp; 0\\
B\  &amp; 0 &amp; 0 &amp; 0 &amp; 10\\
C\  &amp; 4 &amp; 0 &amp; 0 &amp; 5
\end{matrix}&lt;/math&gt; 

which he plugged into 

:&lt;math&gt;\bar{s}^{2}=\left(t_{1}-t_{0}\right)^{2}-\left(x_{1}-x_{0}\right)^{2}-\left(y_{1}-y_{0}\right)^{2}-\left(z_{1}-z_{0}\right)^{2}&lt;/math&gt; 

from which he obtained the sides AB=10, AC=3, CB=3 and the inequality &lt;math&gt;AC+CB&lt;AB&lt;/math&gt;.

|-

|[[w:Arthur Eddington|Eddington]]&lt;ref name=edding2 /&gt;

1922

|He distinguished between the &quot;space-triangle&quot; for spacelike intervals, and the &quot;time-triangle&quot; for time-like intervals. The latter is measured with a clock from A to B and from B to C, with the sum of those readings ''is always less'' than the reading of a clock measuring directly from A to C. In the ordinary space-triangle any two sides are together greater than the third side; in the time-triangle two sides are together ''less'' than the third side. 

|-

|Rogers&lt;ref name=rogers /&gt;

1922

|He showed that the &quot;pure time-triangle&quot; C, A, B (in their proper time order) satisfies the relation &lt;math&gt;\cosh C=\tfrac{\alpha^{2}+\beta^{2}-\gamma^{2}}{2\alpha\beta}&lt;/math&gt;, where &lt;math&gt;\cosh C&lt;/math&gt; denotes the unit-scalar product of the vectors CA, CB, and &lt;math&gt;\alpha,\beta,\gamma &lt;/math&gt; the real and positive intervals BC, CA, AB. Since &lt;math&gt;\alpha&gt;\beta &lt;/math&gt; and &lt;math&gt;\cosh C&gt;1&lt;/math&gt;, it follows that &lt;math&gt;\alpha&gt;\beta+\gamma &lt;/math&gt;. That is, &quot;the greatest side of pure time-triangle is greater than the sum of the other two sides&quot;. It follows at once that the stationary value of the proper time integral is an &quot;absolute maximum&quot;.

|}

==Negligibility of proper acceleration==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|valign=top |[[w:Albert Einstein|Einstein]]

1905-1918


|In 1905,&lt;ref name=einstein05 /&gt; Einstein used velocity time dilation &lt;math&gt;\tau=t\sqrt{1-\left(\frac{v}{V}\right)^{2}}&lt;/math&gt; to derive the retardation of a clock performing a round-trip with constant speed &lt;math&gt;v&lt;/math&gt; along a polygonal path or a continuously curved line, without mentioning any influence of acceleration at turnaround.

{{anchor|Einstein 1911-VA}} In 1911 (published 1912),&lt;ref name=einstein3 /&gt; Einstein said that special relativity doesn't say anything about what happened to the clock's pointer position during the acceleration that changes the clock's direction along the round-trip, yet the influence of this change must be getting smaller the longer the clock ''is moving uniformly'', i.e. the longer one chooses the dimensions of the path.

{{anchor|Einstein 1912-VA}}In an unpublished manuscript on special relativity from 1912,&lt;ref name=einst12manu /&gt; he pointed out that any influence of acceleration during the round-trip experiment, can be neglected if one makes the time of acceleration negligible with respect to the total time of motion along the polygonal path. 

{{anchor|Einstein 1914a-VA}}In a letter from April 1914,&lt;ref name=einstpetz /&gt; Einstein showed that any ''finite'' acceleration at turnaround during the round-trip experiment can only influence the clock in a ''finite'' way, thus it can be neglected by minimizing the time of acceleration with respect to the time of uniform translation. So it ''must be concluded'' that the clock is retarded at reunion after traveling on a polygonal path.

{{anchor|Einstein 1914b-VA}}During a conversation in May 1914,&lt;ref name=rowe group=S /&gt; Einstein is reported to have replied that the accelerations during the round-trip are &quot;irrelevant for the amount of the time difference&quot;. (Compare with {{slink||Einstein 1914b-AC}}) 

{{anchor|Einstein 1918-VA}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from 1918,&lt;ref name=einstein18 /&gt; Einstein pointed out that any effect of velocity changes at turnaround must be limited, thus the traveling clock must be retarded at reunion due to time dilation if one makes the path AB and back along the round-trip long enough. (Compare with {{slink||Einstein 1918-AC}})

|-

|[[w:Emil Wiechert|Wiechert]]&lt;ref name=wiechert11 /&gt;

1911

|{{Anchor|Wiechert 1911-VA}}[[File:WiechertTwin.svg|110px|right]] He demonstrated that differential aging along the round-trip cannot be caused during the passage from one velocity to another (i.e. acceleration) at turnaround, because the same result also follows when ''both'' A and B experience the ''same velocity changes'' with respect to another frame, only with the difference that B has relative velocities &lt;math&gt;+u&lt;/math&gt;  and &lt;math&gt;-u&lt;/math&gt; for a long time, while A is brought after a short time from relative velocity &lt;math&gt;+u&lt;/math&gt; to relative rest at which it remains a long time, and then it is brought to relative velocity &lt;math&gt;-u&lt;/math&gt;  for a short time.

{{Lorentzbox|Text=He was probably the first to use an example in which both accelerate with same magnitude.}}

|-

|[[w:Max von Laue|Laue]]&lt;ref name=laue3 /&gt;

1913

|{{anchor|Laue 1913-VA}}He showed that the problem of the influence of acceleration at turnaround in the round-trip experiment, can be eliminated by ''arbitrarily'' enlarging the time in inertial motion.

{{Lorentzbox|Text=This is the same argument as given in {{slink||Einstein 1911-VA}}. The Einstein-Laue argument was also used by others such as [[w:Hans Thirring|Thirring]] (1921)&lt;ref name=thirring /&gt; or [[w:Max Born|Born]] (1921).&lt;ref name=born /&gt;}}

|-

|[[w:Hendrik Lorentz|Lorentz]]&lt;ref name=lorentz1 /&gt;

1913

|He pointed out that any effect of acceleration on the traveling clock at turnaround, can be separated from the time dilation effect since only the latter depends on the distance traversed along the round-trip.

{{Lorentzbox|Text=Similarly, [[w:Wolfgang Pauli|Pauli]] (1921) stated that the arising infinitesimal accelerations at turnaround are certainly independent of the total travel time and ''therefore easy to eliminate''.&lt;ref name=pauli /&gt;}}
|}

==Relay (three brothers) experiment==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:de:Fritz Grünbaum (Physiker)|Grünbaum]]&lt;ref name=gbaum /&gt;

1911

|He discussed a one-way time dilation experiment in which the first clock is set into motion from the origin and then moving to the second clock. He argued that one can avoid the problem of acceleration experienced by the first clock when set into motion, by replacing it with a ''third'' clock that is already in motion with constant velocity and is synchronized at the origin with the first clock.

{{Lorentzbox|Text=While Grünbaum didn't discuss round-trip experiments, his introduction of a third clock in order to avoid acceleration is the basis of the three-brother experiment.}}

|-

|valign=top|[[w:Emil Wiechert|Wiechert]]

1920-1922

|In 1920 (published 1921),&lt;ref name=wiechert20 /&gt; Wiechert explained how to completely remove acceleration from the round-trip experiment: Bodies A, B, C move undisturbed and non-accelerated in different directions. A and B pass each other at time (1), B and C pass each other at a later time (2), and C and A finally pass each other at an even later time (3). So in this setup, the condition of C is the continuation of the condition of B. On any of the three bodies one can count the oscillations of light of a certain spectral-line, in which case relativity predicts that the ''combined sum of all oscillations'' on B+C is smaller than the number of oscillations on A alone. Wiechert also held that one can replace the light oscillations by the life functions of human-like beings which live on A, B and C. For instance, while the inhabitants of B+C only had time for one meal, there were arbitrarily many generations on A who follow after each other by death and birth.

[[File:Wiechert1922a.png|180px|right]]
In 1921 (published 1922),&lt;ref name=wiechert21 /&gt; Wiechert extended his previous acceleration-free round-trip experiment to an arbitrary number of non-accelerated bodies &lt;math&gt;B_{1}&lt;/math&gt;, &lt;math&gt;B_{2}&lt;/math&gt;, ..., which constitutes a &quot;relay&quot; (German: Stafette) starting from body A and back again. The first B passes A and moves away, and after some time the last B comes back to A. Since any B body continues the fate of the previous one, all bodies &lt;math&gt;B_{1}&lt;/math&gt;, &lt;math&gt;B_{2}&lt;/math&gt;, ..., combined have emitted fewer oscillations than A alone during the relay race. Wiechert pointed out that instead of light oscillations one can also choose the aging of life forms.

{{Lorentzbox|Text=Such relay experiments were later independently rediscovered in English language papers&lt;ref name=debs group=S /&gt; such as by Lange (1927)&lt;ref group=S name=lange /&gt; in which the brothers synchronize their times when they pass each other (“three brother experiment”).}}
|}

==Acceleration as asymmetry indicator==

While it was known that any direct influence of [[w:proper acceleration]] on clocks can be neglected in the computation of the inertial frame of the stay-at-home twin (see previous section {{slink||Negligibility of proper acceleration}}), the very fact that only one of them is accelerating is still useful as an asymmetry argument in order to show that there is no contradiction to the relativity principle.

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Paul Langevin|Langevin]]&lt;ref name=langevin1 /&gt;

1911

|{{Anchor|Langevin 1911-AC}}He derived differential aging in the round-trip experiment using the proper time integral along worldlines (see {{slink||Langevin 1911-PT}}) and used acceleration as an asymmetry indicator: The result of the round-trip experiment is &quot;another example of the absolute character of acceleration&quot; in which the &quot;asymmetry occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity&quot;. 

|-

|[[w:Arnold Sommerfeld|Sommerfeld]]&lt;ref name=sommerfeld /&gt;

1913

|After he showed (see {{slink||Sommerfeld 1913-PT}}) that retardation of time in the round-trip experiment derived from the proper time integral rests on the assumption that the clock's rate ''only depends on its momentary velocity'' (now called &quot;clock hypothesis&quot;), he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle since one of the clocks has to be accelerated in order to come back, thus the retardation in the round-trip experiment does not demonstrate &quot;motion&quot;, but &quot;accelerated motion&quot;.

|-

|[[w:Hendrik Lorentz|Lorentz]]

1913&lt;ref name=lorentz1 /&gt;

|After he derived differential aging in the round-trip experiment from velocity time dilation and pointed out the negligibility of proper acceleration for the computation, he used acceleration as an asymmetry indicator: There is no contradiction to the relativity principle, since one of them changes velocity and accelerates; the relativity principle does not require symmetry between inertial and non-inertial observers.

|-

|valign=top|[[w:Albert Einstein|Einstein]]

1914-1920

|{{anchor|Einstein 1914b-AC}} During a conversation in 1914,&lt;ref name=rowe group=S /&gt; Einstein is reported to have said that moving clock B is retarded because it was accelerating in contrast to clock A; while those accelerations are ''irrelevant'' for the ''amount'' of the time difference, their ''presence'' nevertheless cause B to fall behind (&quot;accelerated motions are absolute&quot;).

{{anchor|Einstein 1918-AC}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from 1918&lt;ref name=einstein18 /&gt;, Einstein pointed out the negligibility of velocity changes from the viewpoint of an inertial frame (see {{slink||Einstein 1918-VA}}). Then he used ''acceleration as an asymmetry indicator'' in order to show, that there is no contradiction to the relativity principle, because relativity only predicts the equivalence of non-accelerated inertial frames: &quot;only K is such a frame while K' is temporarily accelerated, thus the retardation of U2 with respect to U1 cannot be used to construe a contradiction against the theory.&quot; 

{{anchor|Einstein 1920-AC}}Einstein is reported to have said in an interview from 1920:&lt;ref name=einstein20 /&gt;
{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Bei diesen Zwillingen, erklärte Einstein, haben wir zunächst eine ''Gefühls-Paradoxie'' vor uns. Eine ''Denk-Paradoxie'' würde indeß nur dann vorliegen, wenn sich für das Verhalten der beiden Geschöpfe kein zureichender Grund anführen ließe. Dieser Grund für das Jüngerbleiben des A ergibt sich vom Gesichtspunkt der speziellen Relativitätstheorie aus der Tatsache, daß das betreffende Geschöpf — und nur dieses — Beschleunigungen erlitten hat.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |In the case of these two twins,&quot; Einstein declared, &quot;we have merely a paradox of ''feeling''. It would be a paradox of ''thought'' only if no sufficient ground could be suggested for the behaviour of these two creatures . This ground, which counts for the comparative youth of A, is given, from the point of view of the special theory of relativity, by the fact that the creature in question, and only this creature, has been subject to accelerations.&quot;
|}

In a discussion from 1922,&lt;ref name=morand /&gt; Einstein is reported to have said that there is no contradiction in the round-trip experiment (in terms of a train leaving the station and returning later): The relativity principle is not applicable to this case, because the train is not in a Galilean system (i.e. inertial frame) any longer during the period of velocity change at turnaround, i.e. the ensemble of two frames having velocities in opposite direction is not an inertial frame. There is no reciprocity between a frame that changes direction and one that doesn't.

|}

==Frame distribution as asymmetry indicator==

Because any direct influence of proper acceleration on the traveling clock at turnaround can be neglected (see {{slink||Negligibility of proper acceleration}}), the importance of {{slink||Acceleration as asymmetry indicator}} is limited to the mere fact that it reveals that only the traveler was in a non-inertial frame as only he changed his inertial frames, thus instead of emphasizing the occurrence of proper acceleration at turnaround, it's possible to describe the asymmetry more geometrically by emphasizing the different distribution of inertial frames of the twins along their worldlines.

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|valign=top|[[w:Max von Laue|Laue]]

1911-1913

|{{Anchor|Laue 1911/12-VA}} In 1911/12,&lt;ref name=laue1 /&gt; he pointed out that during the time of separation, that clock is most advanced which was at rest in an inertial frame all the time; namely there is ''always one, and only one inertial frame'', in which the locations of separation and re-encounter lie in the same geometric point. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1911/12-PT}}).

In 1912/13,&lt;ref name=laue2 /&gt; he argued that in the round-trip experiment, we indeed can decide, which one of the clocks was steadily at rest in one and the same reference system, and which one was in the meantime at rest in two or more such systems. Among them there is of course a real physical difference. He clarified this fact by alluding to different paths in spacetime (compare with {{slink||Laue 1912/13-PT}}).

In 1913&lt;ref name=laue3 /&gt; Laue pointed out:

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Aber nach unseren Voraussetzungen ruht während der Zeit der Trennung die erste Uhr in ''einem'' berechtigten Bezugssystem, die zweite hingegen ruht zwar sowohl bei der Hin- wie bei der Rückbewegung in berechtigten Bezugssystemen, aber notwendig in ''zwei verschiedenen. Deshalb'' unterscheiden sich beider Schicksale physikalisch. Ließe man die zweite Uhr in der ihr anfangs erteilten Bewegung und schickte man ihr dafür die erste Uhr nach einiger Zeit mit größerer Geschwindigkeit nach, so würde beim Zusammentreffen die erste gegen die zweite zurückgeblieben sein; denn jetzt hat die erste während der Trennung in zwei verschiedenen Systemen geruht. (Footnote: Dem naheliegenden Einwand, daß wir über den Gang einer Uhr während eines Geschwindigkeits''wechsels'' nichts aussagen können, begegnet man am einfachsten mit dem Hinweis, daß man die Zeiten der gleichförmigen Bewegung ''beliebig'' groß gegen die der Beschleunigung machen kann.)

| style=&quot;padding: 0px 20px 0px 20px;&quot; | However, by our presuppositions, one clock is at rest in ''one'' valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in ''two different ones. Therefore'' the two fates differ physically. If we would let remain the second clock in the motion which was given to it at the start, and if we send after it the first clock after some time by a greater velocity, then at the encounter the first one would be retarded with respect to the second one; since now it was the first one that was at rest in two different systems during the separation. (Footnote: The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity ''change'', can be met most simply by the allusion, that we can render the times of uniform motion ''arbitrarily'' great with respect to acceleration..)
|}

|-

|[[w:Werner Bloch|Bloch]]&lt;ref name=bloch /&gt;

September 1918

|{{anchor|Bloch 1918-VA}} He represented the frames with three movable slots K, K' and K”, provided with hooks on which one can hang clocks at the origins of K and K'; while one clock always hangs on a hook of slot K, the other clock moved away with K' and after some time was transferred (neglecting any effect of acceleration) by a mechanical device to slot K” that moves in the other direction, by which it comes back; there is no contradiction to the relativity principle, as one clock rested in one inertial frame while the other one rested in two such frames.

|}

==Perspective of the traveler==

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Paul Langevin|Langevin]]&lt;ref name=langevin1 /&gt;

1911

|{{anchor|Langevin 1911-LI}}[[Image:rstd4.gif|170px|right]] After deriving differential aging from the proper time integral in {{slink||Langevin 1911-PT}} and using human beings in {{slink||Langevin 1911-HU}}, he described the perspectives of both observers using light signals and the Doppler effect. When they separate they see each other live 200 times slower, while at return they see each other live 200 times faster. So ''from the explorer's viewpoint'', in the first year he sees the Earth perform the actions of two days, while in the second year he sees the Earth perform the actions of two centuries. The asymmetry can be seen by noticing, that the observer on Earth in 200 years sees the explorer performs the actions of 1 year. Then the explorer turns around, after which the observer on Earth in 2 days sees the projectile perform the actions of another year.

{{Lorentzbox|Text=Langevin used &lt;math&gt;v=c\left(1-\tfrac{1}{20000}\right)&lt;/math&gt;, producing Lorentz factor &lt;math&gt;\gamma\approx100&lt;/math&gt; and Doppler factor &lt;math&gt;\sqrt{\tfrac{c+v}{c-v}}\approx200&lt;/math&gt;.}}

|-

|[[w:Hendrik Lorentz|Lorentz]]

Lectures published in 1913&lt;ref name=lorentz1 /&gt;

Similar treatment in 1914&lt;ref name=lorentz3 /&gt;

|{{anchor|Lorentz 1913/14-LI}}Described the round-trip experiment in terms of inertial observer A (equipped with clock K) and traveling observer B (equipped with clock K'). In the frame of A, clock K' is retarded with respect to K at reunion due to time dilation. He then described the perspective of the traveling observer B by using two-way propagation of light from K' to K and back to K', leading to three periods defined by the moment of B's turnaround: In the first period the light signals return to K' before turnaround; in the second period the signals are emitted before turnaround and return after turnaround; in the third period emission and return of the signals are both happening after turnaround. Lorentz showed that K is time dilated by a factor of &lt;math&gt;\sqrt{1-v^{2}/c^{2}}&lt;/math&gt; with respect to K' in the first and third period, but in the second period K is ticking ''faster'' than K' by a factor of &lt;math&gt;\sqrt{\tfrac{c+v}{c-v}}&lt;/math&gt; which overcompensates the dilation in the other periods and explains, even from the perspective of B, why K' is retarded with respect to K at reunion.

{{Lorentzbox|Text=In a review of the German translation of Lorentz's book, Einstein (1914) didn't directly mention Lorentz's treatment of the twin paradox, but he wrote that nobody who is seriously interested in relativity should neglect to read that book.&lt;ref name=einstlor /&gt; [[w:Wolfgang Pauli|Pauli]] (1921) refers to Lorentz's book as one of three papers that analyze the twin paradox more closely.&lt;ref name=pauli /&gt;}}

|-

|valign=top| [[w:Albert Einstein|Einstein]]

1916-1920

|{{anchor|Einstein 1916-EP}}In a lecture from 1916,&lt;ref name=einstein16 /&gt; of which only an abstract was published, Einstein spoke about the &quot;clock paradox of special relativity from the standpoint of [[w:general relativity]].&quot;

{{anchor|Einstein 1918a-EP}}In a letter from September 1918,&lt;ref name=einadl /&gt; Einstein showed that general relativity makes the inertial frame K and and the accelerated frame K' of the clocks in the round-trip experiment &quot;equally justified&quot;, explaining the time difference in K' by combining the influence of velocity and gravitational potential on clocks.

{{anchor|Einstein 1918-EP}}In his famous &quot;Dialog about Objections against the Theory of Relativity&quot; from November 1918,&lt;ref name=einstein18 /&gt; aimed at clarifying misconceptions of the clock paradox, he explained that there is no paradox in special relativity because there is no symmetry between clock U1 at rest in inertial frame K and clock U2 at rest in accelerated frame K' (see {{slink||Einstein 1918-AC}}). Yet [[w:general relativity]] and the [[w:equivalence principle]] allow the treatment of this problem also from the standpoint of frame K', where clock U2 remains at rest all of the time while U1 makes the following movements: (1) It is accelerated by a homogeneous gravitational field in the negative direction, (2) it moves with constant velocity &lt;math&gt;-v&lt;/math&gt;, (3) it is accelerated in the positive direction until it turns around and comes by with constant velocity &lt;math&gt;+v&lt;/math&gt;, (4) it moves with velocity &lt;math&gt;+v&lt;/math&gt;, (5) it is accelerated in the negative direction until it stops. Clock U1 is retarded with respect to U2 in periods 2) and 4) due to velocity time dilation, but this retardation is overcompensated by the faster rate of U1 during period 3), because U1 is at a higher gravitational potential. He argued that the computation (which he didn't provide) shows that the advance of U1 in period 3) is double its retardation during periods 2) and 4). Einstein concluded that by this consideration &quot;the paradox is completely resolved&quot;. Using [[w:Mach's principle]], he pointed out that the gravitational field in K' might be induced by the masses of the universe that are accelerated in this frame.

{{anchor|Einstein 1918b-EP}}In a letter to Einstein from December 1918, [[w:Max Jakob|Jakob]] doubted the result that the advance in period 3) is double the retardation during periods 2) and 4). Einstein responded by letter,&lt;ref name=einstein18b /&gt; in which he used the gravitational time dilation factor &lt;math&gt;1+\Phi/c^{2}&lt;/math&gt; in K' in order to show that U1 at distance &lt;math&gt;l&lt;/math&gt; is advancing by &lt;math&gt;\Phi/c^{2}=2vl/c^{2}&lt;/math&gt; in period 3), which is indeed the double of approximated delay &lt;math&gt;vl/c^{2}&lt;/math&gt; caused by velocity time dilation during periods 2) and 4).

{{anchor|Einstein 1921-EP}}Einstein is reported to have said in an interview from 1920,&lt;ref name=einstein20 /&gt; that while acceleration explains the age difference between the stationary twin B and the traveling twin A in terms of special relativity (see {{slink||Einstein 1920-AC}}), the &quot;proper&quot; description in terms of general relativity is as follows:

{| 

! width=50% | German original

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; | Eine tiefere Erfassung des Grundes ist indeß nur auf dem Boden der „Allgemeinen Relativitätstheorie&quot; zu erlangen, die uns erkennen läßt, daß von A aus beurteilt ein Zentrifugalfeld existiert, von B aus betrachtet aber nicht; und dieses Feld hat einen Einfluß auf den relativen Ablauf und die Raschheit der Lebensvorgänge.

| style=&quot;padding: 0px 20px 0px 20px;&quot; | A proper grasp of the reason is furnished only when we adopt the general theory of relativity, which tell us that, from the point of view of A, a centrifugal field exists, whereas it is absent from the point of view of B. This field exerts an influence on the relative rate of happening of the events of life.&quot;
|}

{{Lorentzbox|Text=a) Einstein's explanation was quickly adopted in the textbooks of [[w:Werner Bloch|Bloch]] (1920),&lt;ref name=bloch2 /&gt; [[w:Wolfgang Pauli|Pauli]] (1921),&lt;ref name=pauli /&gt; [[w:August Kopff|Kopff]] (1921),&lt;ref name=kopff /&gt; [[w:Karl Bollert|Bollert]] (1921),&lt;ref name=bollert1 /&gt; [[w:Max Born|Born]] (1921),&lt;ref name=born /&gt; expressing the view that general relativity is &quot;necessary&quot; to provide the &quot;complete&quot; solution of the twin paradox.
b) From a modern standpoint, however, Einstein's explanation has nothing to do with general relativity, but is rather an application of accelerated frames and &quot;pseudo&quot;-gravitational fields to flat Minkowski space of ''special'' relativity.&lt;ref name=weiss group=S /&gt;}}

|-

|[[w:Hans Thirring|Thirring]]&lt;ref name=thirring /&gt;

April 1921

|{{anchor|Thirring 1921-DS}}[[Image:Twin Paradox Minkowski Diagram.svg|right|200px]]
He described the round-trip experiment by using two platforms K (clock A) and K' (clock B) each equipped with rows of clocks. He first demonstrated the symmetry of time dilation and the mutual relativity of simultaneity on the platforms and its effect on clock synchronization. The K clocks that B passes are all advanced because of &lt;math&gt;t'=\gamma\left(t-vx/c^{2}\right)&lt;/math&gt;, and the same is true after turnaround since only the direction of velocity has to be changed in the Lorentz transformation &lt;math&gt;t'-t'_{0}=\gamma\left(t+vx/c^{2}\right)&lt;/math&gt; leading to the effect of clock desynchronization, where &lt;math&gt;t'_{0}&lt;/math&gt; is a constant depending on which clock one uses as standard for the new synchronization. He graphically showed using Minkowski diagrams, that this simultaneity jump due to desynchronization amounts to double the velocity time dilation during the inertial phases, explaining why A is more advanced than B at reunion. 

{{Lorentzbox|Text=Using clock B as synchronization standard, Thirring's constant is given by &lt;math&gt;t'_{0}=2l\gamma v/c^{2}=2t\gamma v^{2}/c^{2}&lt;/math&gt; with &lt;math&gt;l=vt&lt;/math&gt; as position of turnaround. A similar explanation was subsequently given by Langevin (1922).&lt;ref name=morand /&gt;}}

|}

==Curved spacetime==

While the previous examples are defined in flat Minkowski spacetime and therefore can be fully discussed in terms of special relativity, [[general relativity]] is required when [[:w:spacetime curvature]] in the presence of mass and energy cannot be neglected any more.&lt;ref name=koks group=S /&gt; 

{| class=&quot;wikitable&quot; style=&quot;background-color:white;&quot;

|-

! Author !! Early examples

|-

|[[w:Jean Becquerel|Becquerel]]&lt;ref name=becqu1 /&gt;

1922

|After defining gravitational time dilation &lt;math&gt;d\tau=\sqrt{1-\tfrac{2GM}{c^{2}r}}dt&lt;/math&gt; in terms of the [[w:Schwarzschild metric]] around a material center, he discussed the following round-trip experiment: There are two identical clocks A and B placed next to each other, at a point very far from the material center, initially marking the same time &lt;math&gt;t&lt;/math&gt;. Let us transport clock A to a point where the field is more intense, at a distance &lt;math&gt;r&lt;/math&gt; from the center; this clock will measure time &lt;math&gt;\int d\tau&lt;/math&gt; which is shorter than &lt;math&gt;\int dt&lt;/math&gt;, thus it will run more slowly. If we bring clock A back to clock B, we will have to note that it is retarded with respect to B.

|}

==Paradoxical?==

{| class=wikitable style=&quot;background-color:white;&quot;

! width=50% | German original of [[w:Max von Laue|Laue]] (1911/12):&lt;ref name=laue1&gt;Laue introduces the word &quot;paradox&quot;, alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource&lt;/ref&gt;

! English translation

|-

|Unter all den paradox erscheinenden Folgerungen aus der Zeittransformation der Relativitätstheorie gibt es wohl keine, gegen welche sich der natürliche Menschenverstand bei jedem, der der Sache noch ungewohnt ist, so sehr sträubt, wie gegen die, daß die Zeitangabe einer Uhr von ihrem Bewegungszustand abhängen soll. Schon in seiner grundlegenden Arbeit hat Einstein diese Paradoxie auf die Spitze getrieben in einem Gedankenexperiment, welches neuerdings von Langevin in einem auch sonst sehr lesenswerten Vortrage besonders hübsch erläutert worden ist.

|Of all apparently paradox consequences that stem from the time-transformation of the theory of relativity, there is probably none against which the common sense of anyone who is still unfamiliar with the matter is more reluctant, than the one according to which the time indication of a clock shall be dependent on its state of motion. Already in his fundamental paper, Einstein has driven this paradox to the extreme by a thought experiment, recently explained in a very nice way by Langevin in a lecture that is also very readable in other respects.

|-

|colspan=2|{{Lorentzbox|Text=Laue was probably the first to denote the round-trip experiment as paradoxical (even though he pointed out that there are no real contradictions). Subsequently, [[:w:Paul Gruner|Gruner]] (1912)&lt;ref name=gruner /&gt; and others including Einstein (1918)&lt;ref name=einstein18 /&gt; explicitly used the expression &quot;clock paradox&quot; (French: Paradoxe des horloges, German: Uhrenparadoxon), whereas [[w:Rudolf Seeliger|Seeliger]] (1913)&lt;ref name=seel /&gt; spoke of the &quot;familiar Einstein-Langevinian paradox&quot; (German: &quot;bekannte Einstein-Langevinsche Paradoxon&quot;).}}

|}

==Misunderstandings==

{| class=wikitable style=&quot;background-color:white;&quot; 

! width=50% padding=10 | German original by [[w:Otto Berg (scientist)|Berg]] (1910):&lt;ref name=berg /&gt;

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Im Punkte &lt;math&gt;x = 0&lt;/math&gt; des Systems S befinde sich eine Uhr, eine andere im Punkte &lt;math&gt;x'=0&lt;/math&gt; von S'. Diese zweite bewege sich mit S' bis zum Punkte &lt;math&gt;x = a&lt;/math&gt;, kehre dort um und bewege sich nun mit der Geschwindigkeit &lt;math&gt;v&lt;/math&gt; zurück bis zum Punkte &lt;math&gt;x= 0&lt;/math&gt;. Welche Zeit müssen beide Uhren in dem Moment angeben, wo sie sich wieder treffen? Wir beantworten diese Frage zunächst vom Standpunkt des Beobachters in S. Die Uhr in &lt;math&gt;x' = 0&lt;/math&gt; hat sich mit der Geschwindigkeit &lt;math&gt;v&lt;/math&gt; bis zum Punkte &lt;math&gt;x = a&lt;/math&gt; bewegt; dazu brauchte sie die Zeit &lt;math&gt;\tau=\tfrac{a}{v}&lt;/math&gt;. Zum Rückweg ist dieselbe Zeit nötig. Nach der Zeit &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; ist die Uhr also wieder im Punkte &lt;math&gt;x = 0&lt;/math&gt; angelangt. Wir stellen uns nun auf den Standpunkt des Beobachters in S'. Für diesen führt nach dem Relativitätsprinzip das System S genau dieselben Bewegungen aus wie das System S' für den Beobachter in S, nur in entgegengesetzter Richtung. Die Zeit bis zum Zusammentreffen beider Uhren ist also im System S' ebenfalls gegeben durch &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt;. Betrachtungen, die auf anschauliche Vorstellungen, wie Nachgehen von Uhren, gestützt sind, führen hier leicht zu Irrtümern, von denen auch die Fachlitteratur nicht frei ist.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |There is a clock at point &lt;math&gt;x=0&lt;/math&gt; of system S, and another one at point &lt;math&gt;x'=0&lt;/math&gt; of S'. The second one moves together with S' until point &lt;math&gt;x=a&lt;/math&gt;, turns around and now moves back with speed &lt;math&gt;v&lt;/math&gt; to point &lt;math&gt;x=0&lt;/math&gt;. Which time must both clocks indicate at the moment at which they encounter again? We answer this question at first from the standpoint of the observer in S. The clock at &lt;math&gt;x=0&lt;/math&gt; has been moving with speed &lt;math&gt;v&lt;/math&gt; until point &lt;math&gt;x=a&lt;/math&gt;, for which it required time &lt;math&gt;\tau=\tfrac{a}{v}&lt;/math&gt;. The same time is required for the way back. After time &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; the clock has thus arrived again at point &lt;math&gt;x=0&lt;/math&gt;. Let's now take the standpoint of the observer in S'. In his view in accordance with the relativity principle, system S is conducting exactly the same motions as those of system S' with respect to the observer in S, only in opposite direction. Thus the time until the meeting of both clocks is given by &lt;math&gt;2\tau=2\tfrac{a}{v}&lt;/math&gt; in system S' as well. Considerations based on illustrative notions, such as the retardation of clocks, easily lead to mistakes at this place, of which also the professional literature isn't free.

|-

|colspan=2|{{Lorentzbox|Text=Berg was probably the first to turn the relativity principle against asymmetric aging in the round-trip experiment, claiming that both clocks must indicate the same time at reunion. See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}}

|-

! width=50% | German original by [[w:Emil Wiechert|Wiechert]] (1911)&lt;ref name=wiechert11 /&gt;

! English translation

|-

|colspan=2| Even though he correctly derived differential clock aging in the round-trip experiment, he claimed that effects like time dilation are &quot;apparent&quot; if one admits Einstein's &quot;unconditional&quot; relativity principle in which there is no aether and all &quot;strides&quot; (i.e. non-accelerated motions) are physically equivalent, but they are &quot;real&quot; if one admits the existence of an aether in the framework of a &quot;conditional&quot; relativity principle in which all strides are physically non-equivalent or anisotropic. This led him to the following interpretation of the clock paradox:

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |[...] so muß am Schluß des Versuches B in seinem Fortschritt gegenüber A im Verhältnis &lt;math&gt;1:\sqrt{1-u^{2}/c^{2}}&lt;/math&gt; zurückgeblieben sein. Und dieses Zurückbleiben ist unbedingt reell, denn die beiden Gebilde A und B können ja unter gleichen Umständen unmittelbar beieinander verglichen werden. Hier ist es ganz sicher ausgeschlossen, an einen Schein zu glauben, der durch unsere Auffassung der Zeit bewirkt wird. So ist denn also auch die Folgerung unabwendbar, daß für den Verlauf der Weltvorgänge die Schreitungen nicht gleichwertig sind, ''und damit sind wir von neuem zu einem Schluß gekommen, welcher der Unbedingtheit des Relativitätsprinzipes durchaus widerspricht.'' [...] Man kann den Versuch noch mannigfach variieren, z. B. so, daß A ebenso wie B zwei verschiedene Schreitungen, &lt;math&gt;+u&lt;/math&gt; und &lt;math&gt;-u&lt;/math&gt;, nacheinander inne hat. Wird dann zu A der Wert &lt;math&gt;u_{1}&lt;/math&gt;, zu B  der Wert &lt;math&gt;u_{2}&lt;/math&gt;, zugeordnet, so muß der Vergleich von A  und B am Schluß des Versuches ergeben, daß B  oder A in seinem Fortschritt zurückgeblieben erscheint, je nachdem die Schreitungen &lt;math&gt;+u_{1}&lt;/math&gt;, &lt;math&gt;-u_{1}&lt;/math&gt;, oder &lt;math&gt;+u_{2}&lt;/math&gt;, &lt;math&gt;-u_{2}&lt;/math&gt; weiter auseinanderliegen. ''Vielleicht ist gerade diese Formulierung des Satzes besonders geeignet, um die Ungleichwertigkeit der verschiedenen Schreitungen klar und deutlich zu zeigen.''

| style=&quot;padding: 0px 20px 0px 20px;&quot; | [...] thus B's progress must be retarded with respect to A's in the ratio &lt;math&gt;1:\sqrt{1-u^{2}/c^{2}}&lt;/math&gt; at the end of the experiment. And this retardation is definitely real, since both bodies A and B indeed can be immediately compared side by side under the same conditions. Here it is certainly excluded to believe that this is an appearance due to our conception of time. Thus the consequence is unavoidable too, that the strides are not equivalent in the course of the world processes, ''and therefore we again came to a conclusion that completely contradicts the unconditionality of the relativity principle.'' [...] One can vary this experiment in many ways, for instance, so that A in the same way as B successively undergoes two different strides &lt;math&gt;+u&lt;/math&gt; and &lt;math&gt;-u&lt;/math&gt;. If we apply the value &lt;math&gt;u_{1}&lt;/math&gt; to A and &lt;math&gt;u_{2}&lt;/math&gt; to B, then the comparison of A and B at the end of the experiment must give the result, that B or A is retarded in its progress depending on whether the strides &lt;math&gt;+u_{2},-u_{2}&lt;/math&gt; or &lt;math&gt;+u_{1},-u_{1}&lt;/math&gt; are further apart. ''Probably it is precisely this formulation of the theorem that is particularly suitable to demonstrate the non-equivalence of the different strides clearly and explicitly.''

|-

|colspan=2|{{Lorentzbox|Text=This interpretation was directly rebutted by Laue (1911/12) who demonstrated the geometrical meaning of differential aging in Minkowski space, see sections {{slink||Laue 1911/12-PT|Laue 1911/12-VA}}, showing that there is no need to assume non-equivalance or anisotropy of motions. Laue added, that as long as there is no experimental contradiction to the relativity principle, the question after the aether can be banned from physics and left to philosophy.&lt;ref name=laue1 /&gt;}}

|-

! width=50% | German original by [[w:Norman Robert Campbell|Campbell]] (November 1911, published 1912)&lt;ref name=camp /&gt;

! English translation

|-

|colspan=2|After describing the round-trip experiment (as given by Wiechert) according to which the traveling clock B is retarded when it returns with respect to stationary clock A, he abandoned differential clock aging as follows: 

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |Dieser Schluß ist nicht richtig. Die Beziehung zwischen &lt;math&gt;t&lt;/math&gt;, der Ablesung an der Uhr auf A seitens des Beobachters auf A und &lt;math&gt;t'&lt;/math&gt;, der Ablesung an der Uhr auf B seitens des Beobachters auf A, ist (unter der Annahme, daß zu Beginn des Versuchs &lt;math&gt;t=t'&lt;/math&gt; ist)
:&lt;math&gt;t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)&lt;/math&gt;. 
Der Unterschied zwischen &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; ist eine Funktion von &lt;math&gt;z&lt;/math&gt; und &lt;math&gt;v&lt;/math&gt; allein. Wenn man diesen Größen ihre früheren Werte wiedergibt, indem man die beiden Uhren wieder zur Koinzidenz bringt, während sie relativ zueinander ruhen, so geht der Unterschied zwischen &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; wieder auf null zurück, gleichviel, welche Werte &lt;math&gt;z&lt;/math&gt; und &lt;math&gt;v&lt;/math&gt; während der Zwischenzeit gehabt haben mögen. Wenn an irgendeinem Punkte der Bahn die Geschwindigkeit von B relativ zu A eine endliche plötzliche Änderung erfährt, so erfährt auch der Wert von &lt;math&gt;v&lt;/math&gt; eine endliche plötzliche Änderung.

| style=&quot;padding: 0px 20px 0px 20px;&quot; |This conclusion is not correct. The relationship between &lt;math&gt;t&lt;/math&gt; as the reading on the clock on A by the observer on A, and &lt;math&gt;t'&lt;/math&gt; as the reading on the clock on B by the observer on A, is given by (assuming that &lt;math&gt;t=t'&lt;/math&gt; at the beginning of the experiment)
:&lt;math&gt;t'=\frac{1}{\sqrt{1-v^{2}/c^{2}}}\left(t-vz/c^{2}\right)&lt;/math&gt;. 
The difference between &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; is a function of &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;v&lt;/math&gt; alone. If these quantities are given their previous values by bringing the two clocks back to coincidence during which they are at rest relative to one another, the difference between &lt;math&gt;t'&lt;/math&gt; and &lt;math&gt;t&lt;/math&gt; goes back to zero, no matter what values &lt;math&gt;z&lt;/math&gt; and &lt;math&gt;v&lt;/math&gt; may have had in the meantime. If at any point on the path the speed of B experiences a finite sudden change relative to A, then the value of &lt;math&gt;t'&lt;/math&gt; also undergoes a finite sudden change.

|-

|colspan=2|{{Lorentzbox|Text=So Campbell claims that any time difference during the outbound path is wiped out during the inbound path. His mistake is obvious: Campbell is confusing coordinate differences stemming from the Lorentz transformation of ''events'' (which indeed depend on position and direction) with differences in ''clock aging'' derived from the proper time integral (which is ''accumulative'' and independent of position and direction.)}}

|-

! width=50% | French original by [[w:Paul Gruner|Gruner]] (March 1912):&lt;ref name=gruner /&gt;

! English translation

|-

| style=&quot;padding: 0px 20px 0px 20px;&quot; |[...] deux personnes du même âge, se séparant dans des systèmes de « marche » très différents et retournant après un laps de temps assez long, constateront une différence d'âge très sensible. [...] le principe de relativité exige toujours la ''réciprocité parfaite'' des phénomènes entre deux systèmes qui possèdent un mouvement relatif. Si, dans l'exemple cité, les deux personnes du même âge se séparent avec une vitesse relative pour se retrouver plus tard, la constatation d'une différence d'âge sera parfaitement mutuelle : A dira positivement que B est resté en arrière dans son développement, et B affirmera avec le même droit que c'est A qui ne s'est pas développé assez vite. Ainsi le principe absolu de la relativité montre ses conséquences les plus extrèmes et il est clair que l'introduction de l’éther n'est plus en état de résoudre cette contradiction irréductible et inconcevable.

| style=&quot;padding: 0px 20px 0px 20px;&quot; | [...] two people of same age, separating into very different systems of motion and returning after a quite long period of time, will notice a very significant age difference. [...] the principle of relativity always requires the ''perfect reciprocity'' of the phenomenons between two systems that possess relative motion. When, in the cited example, the two persons of same age are separated by some relative velocity only to meet again later, the finding of an age difference will be perfectly mutual: A will positively say that B stayed behind in its development, and B will assert with same right that it was A who has not developed fast enough. By that, the absolute relativity principle shows its most extreme consequences and it is clear, that the introduction of the aether is no longer able to resolve this irreducible and inconceivable contradiction.

|-

|colspan=2|{{Lorentzbox|Text=Gruner was probably the first to claim that combining the round-trip experiment with the symmetry of time dilation leads to the contradictory situation, that both must attribute younger age to one another at reunion. At the end of his paper, we also find the expression &quot;clock paradox&quot; (French: paradoxe des horloges). See [[w:Twin paradox]] as well as sections {{slink||Acceleration as asymmetry indicator|Frame distribution as asymmetry indicator|Perspective of the traveler}} for the solution of that problem.}}

|}

==Historical references==

&lt;references&gt;

&lt;ref name=einstein05&gt;See p. 904f in: {{Citation |author=Einstein, A. |date=1905 |title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik |volume=322 |issue=10 |pages=891–921 |doi=10.1002/andp.19053221004|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 2, Document 23}}. See also: [https://www.fourmilab.ch/etexts/einstein/specrel/www/ English translation at fourmilab].&lt;/ref&gt;

&lt;ref name=einstein11a&gt;See p. 10. in: {{Citation |author=Einstein, A. |title=Die Relativitäts-Theorie|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |issue=1-2|pages=1–14 |date=27 November 1911|orig-date=Lecture 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n11/mode/2up|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 3, Document 17}}.&lt;br /&gt; The publication date 27 November 1911 can be seen on the [https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n5/mode/2up Title page and TOC of issue 1-2].&lt;/ref&gt;

&lt;ref name=einstein3&gt;Discussion between Einstien, Müller, Lämmel and others after the Zürich lecture: {{Citation |author=Einstein, A.; Müller, F., Lämmel, R.|title=Diskussion zu &quot;Die Relativitäts-Theorie&quot;|journal=Naturforschende Gesellschaft, Zürich, Vierteljahresschrift |volume=56 |pages=II-IX |date=January 1912|orig-date=Lecture on 16 January 1911|url=https://archive.org/details/naturforschendegesellschaftinzurich_vierteljahrsschriftdernaturforschendengesellschaftinzur_v56_1911/page/n587/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 3, Document 18, and in the corresponding English translation volume}}&lt;br /&gt; While the discussion already happened on January 1911, the publication followed one year later in January 1912 in the session proceedings (Sitzungsberichte) of the third issue, see [https://www.ngzh.ch/publikationen/vjs/56/3 Full issue Nr. 3] with [http://www.ngzh.ch/archiv/1911_56/56_1-2/56_3.pdf Title page and TOC] and the [http://www.ngzh.ch/archiv/1911_56/56_3/56_30.pdf Sitzungsberichte including Einstein's discussion on pp. II-IX]. &lt;/ref&gt;

&lt;ref name=einst12manu&gt;See p. 46 in: {{Citation |author=Einstein, A. |date=1912 |chapter=Document 1: Einstein's manuscript on the special theory of relativity|title=The collected papers of Albert Einstein|volume=4|pages=3-108|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstlor&gt;{{Citation|author=Einstein, A.|date=1914|title=Review of &quot;Lorentz, H. A. – Das Relativitätsprinzip&quot; |journal=Die Naturwissenschaften|volume=2|pages=1018|url=https://archive.org/details/CAT31421305002/page/1018/mode/2up|quote=Reprinted in  ''The Collected Papers of Albert Einstein'', Vol. 6, Document 11}}&lt;/ref&gt;

&lt;ref name=einstpetz&gt;{{Citation |author=Einstein, A. |date=1914 |chapter=Document 5: Letter from Einstein to Petzoldt|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstein16&gt;See p. 423f in: {{Citation |author=Einstein, A. |date=1916 |title=Announcement of Einstein's lecture &quot;Über einige anschauliche Überlegungen aus dem Gebiete der Relativitätstheorie&quot;|journal=Berliner Sitzungsberichte|pages=423|volume=1916 (part 1)|url=https://archive.org/details/sitzungsberichte1916deutsch/page/423/mode/2up}}&lt;/ref&gt;

&lt;ref name=einadl&gt;Letter exchange between Einstein and Adler in which the critique on the clock paradox by Berg (1910) and Petzoldt (1914) was mentioned, together with the general relativity solution in terms of the gravitational potential, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Adler's letter in Document 620 and Einstein's reply in Document 628|title=The collected papers of Albert Einstein|volume=8a|pages=16-17|trans-chapter=See also the English translation in the corresponding translation volume}}&lt;/ref&gt;

&lt;ref name=einstein18&gt;Einstein discussed in terms of inertial frames (special relativity) on pp. 697f; accelerated frames (general relativity) on pp. 698f.; distant masses (Mach's principle) on pp. 700f. in: {{citation |author=Einstein, A.|title=Dialog über Einwände gegen die Relativitätstheorie|date=November 1918|volume=6|issue=48|journal=Die Naturwissenschaften|pages=697-702|url=https://archive.org/details/sim_naturwissenschaften_1918-11-29_6_48|quote=Reprinted in ''The Collected Papers of Albert Einstein'', Vol. 7, Document 13}}; See also English translation [[:s:Translation:Dialog about Objections against the Theory of Relativity|Dialog about Objections against the Theory of Relativity]] on Wikisource.&lt;/ref&gt;

&lt;ref name=einstein18b&gt;Letter exchange between Max Jakob and Einstein from December 1918, in: {{Citation |author=Einstein, A. |date=1918 |chapter=Jakob's letter in Document 661c and Einstein's reply in Document 663a|title=The collected papers of Albert Einstein|volume=10|pages=189-190}}&lt;/ref&gt;

&lt;ref name=einstein20&gt;Interview of Einstein by Moszkowski, see p. 204f. in: {{citation |author=Moszkowski, A.|title=Einstein. Einblicke in seine Gedankenwelt|orig-date=Copyright date 1920 |date=1921|place=Hamburg|url=https://www.archive.org/details/einsteineinblick00moszuoft}}; See also English translation by H. L. [[Henry Brose|Brose]] (1921): [https://archive.org/details/einsteinsearch00moszrich Einstein, the searcher], p. 206&lt;/ref&gt;

&lt;ref name=morand&gt;Discussion between Painlevé, Einstein, and Langevin on p. 316ff in: {{citation |author=Morand, M.|title=Einstein au collège de france|date=April 1922|journal=La Nature|volume=50|issue=2511|pages=315-320|url=http://cnum.cnam.fr/CGI/fpage.cgi?4KY28.102/319/100/620/5/613}}&lt;/ref&gt;

&lt;ref name=lammel&gt;{{Citation|author=Lämmel, R.|date=28 April 1911|title=Die Relativitäts-Lehre|journal=Neue Zürcher Zeitung|volume=117|pages=1|url=https://www.e-newspaperarchives.ch/?a=d&amp;d=NZZ19110428-01.2.4.1}}; English translation of the part concering the twin pardox at [[:v:History of Topics in Special Relativity/Twin paradox#Lämmel 1911-Hum|Wikiversity:Early history of the twin paradox - Lämmel]]&lt;/ref&gt;

&lt;ref name=lammel2&gt;See p. 84ff in: {{Citation|author=Lämmel, R.|date=1921|orig-date=Preface December 1920|title=Die Grundlagen der Relativitätstheorie|place=Berlin|publisher=Springer|url=https://archive.org/details/diegrundlagende00lmgoog}}&lt;/ref&gt;

&lt;ref name=langevin1&gt;He derived differential aging from the proper time integral; pointed out that this demonstrates the &quot;absolute nature of acceleration&quot; with respect to an aether, see: {{citation |author=Langevin, P.|title=[[:s:fr:L’Évolution de l’espace et du temps|L’Évolution de l’espace et du temps]]|journal=Scientia |volume=X |pages=31–54 |date=July 1911|orig-date=Lecture 10 April 1911}}; English translation [[:s:en:Translation:The Evolution of Space and Time|The Evolution of Space and Time]] on Wikisource&lt;/ref&gt;

&lt;ref name=langevin2&gt;See p. 329 in: {{citation |author=Langevin, P. |title=Le temps, l'espace et la causalité dans la physique moderne |journal=Bulletin de la Société française de philosophie |volume=12 |orig-date=Lecture October 1911|date=1912|pages=1-28|url=http://ahp.li/1f7fc22d283fdf0deeca.pdf}}&lt;/ref&gt;

&lt;ref name=wiechert11&gt;See p. 745f. general description and proper time; 757f. space travel; in: {{Citation |author=Wiechert, E. |date=September 1911|orig-date=Lectures March-May 1911, submitted 26 July|title=[[:s:de:Relativitätsprinzip und Äther|Relativitätsprinzip und Äther]]|journal=Physikalische Zeitschrift |volume=12 |issue=17-18 |pages=[https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/741 689-707] published September 1; [https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/789 737–758] published September 15}}&lt;/ref&gt;

&lt;ref name=wiechert15&gt;See p. 46 (Einstein, Langevin, Wiechert) and pp. 51f (Laue versus Wiechert) in: {{citation |author=Wiechert, E.|contribution=Die Mechanik im Rahmen der allgemeinen Physik| title=Die Kultur der Gegenwart: Physik|volume=3.3.1|date=1915 |orig-date=Submitted July 1914|pages=1–78|contribution-url=https://www.archive.org/details/physikunterredak00warbuoft}}&lt;/ref&gt;

&lt;ref name=wiechert20&gt;See p. 46f in: {{citation |author=Wiechert, E.|title=Der Äther im Weltbild der Physik|orig-date=Presented December 1920|date=1921|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse|pages=29-70|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN00250586X}}&lt;/ref&gt;

&lt;ref name=wiechert21&gt;See p. 25ff in: {{citation |author=Wiechert, E.|title=[[:s:de:Prinzipielles über Äther und Relativität|Prinzipielles über Äther und Relativität]]|date=1922|orig-date=Lecture September 1921|journal=Physikalische Zeitschrift|volume=23|pages=25-28}}&lt;/ref&gt;

&lt;ref name=muller&gt;See p. 9 in: {{Citation|author=Müller, F.|date=October 1911|journal=Berliner Tageblatt|title=[[:s:de:Das Zeitproblem (1911)|Das Zeitproblem]]|pages=[https://www.deutsche-digitale-bibliothek.de/newspaper/item/2QKOIOLGNVQILTCEZQOGQPLTRVLPM5PZ?query=zeit&amp;issuepage=9 Part 1 published 16 October 1911] and [https://www.deutsche-digitale-bibliothek.de/newspaper/item/IO44I6QBC4SVV5YUKUDSGXYIPQUXXBN5?query=zeit&amp;issuepage=11 Part 2 published 23 October 1911]}}&lt;/ref&gt;

&lt;ref name=gruner&gt;See p. 253f in: {{Citation |author=Gruner, P. |title=[[:s:fr:Rapport sur la dernière discussion concernant le principe de la relativité et l’éther|Rapport sur la dernière discussion concernant le principe de la relativité et l’éther]] |journal=Archives des sciences physiques et naturelles |volume=33|issue=4 |pages=252-254 |date=March 1912}}&lt;/ref&gt;

&lt;ref name=laue3&gt;See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource&lt;/ref&gt;

&lt;ref name=weyl&gt;See p. 147f. in: {{Citation |author=Weyl, H. |date=March 1918|title=Raum-Zeit-Materie (first edition)|publisher=Berlin: Springer|url=https://archive.org/details/RaumZeitMaterieVolIMeinerFrauGewidmet}}; English translation of the 4th edition by H. [[Henry Brose|Brose]] (1921): [https://www.gutenberg.org/ebooks/43006 Space—Time—Matter], pp. 278f.&lt;/ref&gt;

&lt;ref name=gbaum&gt;See footnote on p. 507 in: {{Citation|author=Grünbaum, F. |title=Über einige ideelle Versuche zum Relativitätsprinzip|journal=Physikalische Zeitschrift|volume=12|pages=500–509|date=1911|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0012/page/540}}&lt;/ref&gt;

&lt;ref name=laue1&gt;Laue introduces the word &quot;paradox&quot;, alludes to Berg and discusses Wiechert, in: {{citation |author=Laue, M. v. |title=Zwei Einwände gegen die Relativitätstheorie und ihre Widerlegung |journal=Physikalische Zeitschrift |volume=13 |issue=3|date=February 1912|orig-date=Submitted December 1911|pages=118–120|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/148}}; {{icon|wikisource}} See also English translation [[:s:Translation:Two Objections Against the Theory of Relativity and their Refutation|Two Objections Against the Theory of Relativity and their Refutation]] on Wikisource&lt;/ref&gt;

&lt;ref name=laue2&gt;See p. 42f. for general description; p. 58f. in terms of proper time; in: {{Citation |author=Laue, M. v. |orig-date=Preface December 1912|date=1913 |title=Das Relativitätsprinzip (Second Edition) |publisher=Vieweg |place=Braunschweig|url=https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE4597082}}; See also English translation [[:s:Translation:The Principle of Relativity (Laue 1913)|The Principle of Relativity, Second edition, Part III]] on Wikisource&lt;/ref&gt;

&lt;ref name=laue3&gt;See p. 113f in: {{citation |author=Laue, M. v. |title=Das Relativitätsprinzip |journal=Jahrbücher der Philosophie |volume=1 |date=1913 |pages=99–128}}; {{icon|wikisource}} See also English translation of [[:s:Translation:The Principle of Relativity (Laue, Philosophy)|The Principle of Relativity]] on Wikisource&lt;/ref&gt;

&lt;ref name=berg&gt;See p. 369f in: {{Citation |author=Berg, O. |date=1910 |title=Das Relativitätsprinzip der Elektrodynamik |journal=Abhandlungen der Fries'schen Schule |volume=3 |issue=2|pages=333-382 |url=http://hdl.handle.net/2027/hvd.hnuynk?urlappend=%3Bseq=351}}&lt;/ref&gt;

&lt;ref name=camp&gt;See p. 123f in: {{Citation |author=Campbell, N. |title=Relativitätsprinzip und Äther: Eine Entgegnung an Herrn Wiechert |journal=Physikalische Zeitschrift |volume=13 |pages=120-128 |issue=3|orig-date=Submitted December 1911|date=February 1912|url=https://resolver.sub.uni-hamburg.de/kitodo/PPN891110208_0013/page/150}}. The is based on an English manuscript translated by Max Iklé, and Campbell's first name was Germanised as &quot;Normann&quot;.&lt;/ref&gt;

&lt;ref name=seel&gt;{{Citation|author=Seeliger, R.|title=Review of &quot;P. Gruner – Rapport sur la dernière discussion concernant le principe de la relativité et l'éther&quot;|journal=Die Fortschritte der Physik|volume=68|issue=2|pages=336|date=1913|url=https://books.google.com/books?id=fSJGAQAAMAAJ&amp;pg=PA336}}&lt;/ref&gt;

&lt;ref name=study&gt;See footnote on p. 111 in: {{citation |author=Study, E. |title=Vorlesungen über ausgewählte Gegenstände der Geometrie |date=June 1911|url=https://archive.org/details/vorlesungenber00studuoft|publisher=B.G. Teubner|place=Leipzig}} &lt;/ref&gt;

&lt;ref name=robb1&gt;See pp. 356ff. in: {{Citation|author=Robb, A.|date=1914|title=A theory of time and space|place=Cambridge|publisher=University Press|url=https://archive.org/details/theoryoftimespac00robbrich}} &lt;/ref&gt;

&lt;ref name=robb2&gt;See §12 in: {{citation |author=Robb, A. A.|title=The Straight Path|date=1920 |journal=Nature|pages=599|volume=104|issue=2623|url=https://archive.org/details/sim_nature-uk_1920-02-05_104_2623/page/598/mode/2up}}&lt;/ref&gt;

&lt;ref name=edding2&gt;See p. 22 in: {{Citation |author=Eddington, A. S. |date=1922 |title=The theory of relativity, and its influence on scientific thought |publisher=Oxford Clarendon Press |url=https://archive.org/details/cu31924005748573}}&lt;/ref&gt;

&lt;ref name=rogers&gt;{{citation |author=Rogers, R. A. P.|title=The Time-Triangle and Time-Triad in Special Relativity|date=November 1922|journal=Nature|volume=110|issue=2769|pages=698–699|url=https://archive.org/details/sim_nature-uk_1922-11-25_110_2769/page/698/mode/2up}}&lt;/ref&gt;

&lt;ref name=lorentz1&gt;See pp. 37f, 55ff in: {{citation |author=Lorentz, H. A.|date=1913|title=Het relativiteitsbeginsel : drie voordrachten gehouden in Teyler's stichting|publisher=De Erven Loosjes |place=Haarlem|url=https://resolver.kb.nl/resolve?urn=MMKB24:063387000:00005}}; German translation on pp. 31f, 47f in: {{citation |author=Lorentz, H. A.|date=1914| title=Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|publisher=B.G. Teubner |place=Leipzig and Berlin|url=https://archive.org/details/bub_gb_89PPAAAAMAAJ}}; See also the transcription [[:s:de:Das Relativitätsprinzip (Lorentz)|Das Relativitätsprinzip]] on German Wikisource and the English translation [[:s:Translation:The Principle of Relativity (Lorentz)|The Principle of Relativity]] on English Wikisource&lt;/ref&gt;

&lt;ref name=lorentz3&gt;See §12 in: {{citation |author=Lorentz, H. A.|title=Considérations élémentaires sur le principe de relativité|date=1914 |journal=Revue générale des sciences pures et appliquées|pages=179-186|url=https://archive.org/details/revuegnraled25pari/page/178/mode/2up}}&lt;/ref&gt;

&lt;ref name=bloch&gt;See pp. 67 ff. in: {{Citation | author=Bloch, W.| date=September 1918|title=Einführung in die Relativitätstheorie| publisher=B. G. Teubner |url=https://hdl.handle.net/2027/njp.32101040276907}}&lt;/ref&gt;

&lt;ref name=bloch2&gt;See pp. 69ff. (special relativity) and 102ff. (general relativity) in: {{Citation | author=Bloch, W.| date=1920 |title=Einführung in die Relativitätstheorie (second edition)| publisher=B. G. Teubner |url=https://www.archive.org/details/einfhrungindier00blocgoog}}&lt;/ref&gt;

&lt;ref name=bollert1&gt;See p. 6 (special relativity), pp. 24-26 (EP) in: {{citation |author=Bollert, K.|title=Einstein’s Relativitätstheorie und ihre Stellung im System der Gesamterfahrung |date=April 1921|publisher=Steinkopff|url=https://archive.org/details/dbc.wroc.pl.001504}}&lt;/ref&gt;

&lt;ref name=born&gt;See pp. 190f. (special relativity), 250f (EP) in: {{Citation | author=Born, M.| date=1921 |title=Die Relativitätstheorie Einsteins und ihre physikalischen Grundlagen (Second edition)| publisher=Springer | place=Berlin|url=https://hdl.handle.net/2027/mdp.39015017387310}}; The [https://preserver.beic.it/delivery/DeliveryManagerServlet?dps_pid=IE5426498 first edition (1920)] of Born's book didn't include the twin paradox. English translation of the third edition by H. Brose (1924): [https://archive.org/details/einsteinstheoryo00born Einstein's theory of relativity]&lt;/ref&gt;

&lt;ref name=pauli&gt;See p. 558f (general description); p. 624f (proper time); p. 713f (accelerated frames); in: {{Citation |author=Pauli, W. |date=1921 |journal=Encyclopädie der Mathematischen Wissenschaften|title=Die Relativitätstheorie|pages=539–776|volume=5|issue=2 |url=http://resolver.sub.uni-goettingen.de/purl?PPN360709672}}; English translation by G. Field (1958): [https://books.google.com/books?id=rc3DAgAAQBAJ Theory of Relativity]&lt;/ref&gt;

&lt;ref name=thirring&gt;See p. 209ff in: {{citation |author=Thirring, H.|title=Über das Uhrenparadoxon in der Relativitätstheorie|date=April 1921|journal=Naturwissenschaften|volume=9|issue=18|pages=209-212|url=https://archive.org/details/sim_naturwissenschaften_1921-04-01_9_13/mode/2up}}&lt;/ref&gt;

&lt;ref name=sommerfeld&gt;See p. 71 in: {{citation |author=Sommerfeld, A. |date=May 1913|chapter=Remarks on Minkowski's &quot;Space and Time&quot;|title=Das Relativitätsprinzip|editor=Otto Blumenthal|pages=69-73|url=https://www.archive.org/details/dasrelativittsp00minkgoog}}&lt;/ref&gt;

&lt;ref name=kopff&gt;See pp. 45ff (special relativity and proper time); pp. 117ff (EP); pp. 189ff (Mach's principle), in: {{citation |author=Kopff, A.|title=Grundzüge der Einsteinschen Relativitätstheorie |date=February 1921|publisher=S. Hirzel|place=Leipzig|url=https://www.archive.org/details/grundzgedereins00kopfgoog}}; English translation by H. Levy (1923): [https://hdl.handle.net/2027/mdp.39015017188817 The mathematical theory of relativity].&lt;/ref&gt;

&lt;ref name=becqu1&gt;See p. 48ff (proper time), p. 240f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Le Principe de relativité et la théorie de la gravitation|Le Principe de relativité et la théorie de la gravitation]] |date=1922 |publisher=Gauthier-Villars|place=Paris}}; See also p. 57ff (proper time), p. 177f (general relativity) in: {{citation |author=Becquerel, J.|title=[[:s:fr:Exposé élémentaire de la théorie d’Einstein et de sa généralisation|Exposé élémentaire de la théorie d’Einstein et de sa généralisation]]|date=1922 |publisher=Payot|place=Paris}}&lt;/ref&gt;

&lt;/references&gt;

==Secondary sources==

&lt;references group=S&gt;

&lt;ref name=miller&gt;{{Citation |author=Miller, A. I. |date=1981 |title=Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911) |place=Reading |publisher=Addison–Wesley |isbn=978-0-201-04679-3}}; See section 7.4.13 (Langevin, Wiechert, Laue, Einstein), footnotes 29-34 of chapter 7 (Petzoldt, Sommerfeld, Bergson, Einstein)&lt;/ref&gt;

&lt;ref name=lange&gt;{{Citation|author=Lange, L.|date=1927|title=The clock paradox of the theory of relativity|journal=The American Mathematical Monthly|volume=34|issue=1|pages=22-30|jstor=2299914}}&lt;/ref&gt;

&lt;ref name=pes&gt;{{Citation |author=Pesic, P. |date=2003 |title=Einstein and the twin paradox |journal=European Journal of Physics |volume=24 |issue=6 |pages=585–590 |doi=10.1088/0143-0807/24/6/004}}&lt;/ref&gt;

&lt;ref name=during&gt;{{Citation |author=During, É. |date=2014 |title=Langevin ou le paradoxe introuvable |journal=Revue de métaphysique et de morale |volume=84 |pages=513-527 |doi=10.3917/rmm.144.0513|doi-access=free}}; See pp. 515f (Langevin), 520f. (Einstein, Laue, Weyl, Painlevé).&lt;/ref&gt;

&lt;ref name=debs&gt;{{Citation |author=Debs, T. A., &amp; Redhead, M. L. |title=The twin paradox and the conventionality of simultaneity |date=1996 |journal=American Journal of Physics |volume=64|issue=1| pages=384-392 |doi=10.1119/1.18252}}&lt;/ref&gt;

&lt;ref name=alizzi&gt;{{Citation |author=Alizzi, A., Sen, A., &amp; Silagadze, Z. K.|title=Do moving clocks slow down? |year=2022 |journal=European Journal of Physics |volume=43|issue=6|pages=065601 |doi=10.1088/1361-6404/ac93ca|arxiv=2209.12654}}; Appendix B with reference to Lange and Halsbury&lt;/ref&gt;

&lt;ref name=beng&gt;{{Citation |author=Benguigui, L. G. |date=2020 |title=A Tale Of Two Twins: The Langevin Experiment Of A Traveler To A Star |publisher=World Scientific|isbn=9789811219115}}; See early solutions (Einstein, Langevin, Lorentz, Born/Kopff) and the Bergson controversy. A shorter version appeared in {{arxiv|1212.4414}}.&lt;/ref&gt;

&lt;ref name=rowe&gt;{{Citation|author=Rowe, D. E.|date=2006|title=Einstein's allies and enemies: Debating relativity in Germany 1916–1920|journal=Interactions: Mathematics, Physics and Philosophy|pages=231-280|publisher=Springer|doi=10.1007/978-1-4020-5195-1_8}}; Covering the criticism of Gehrcke starting with 1912; discussion between Einstein and Gehrcke in 1914; Einstein's dialogue (1918) as response to antirelativists; the Weyland event in 1920 and Einstein's response.&lt;/ref&gt;

&lt;ref name=weiss&gt;Weiss, W. (Physics FAQ): [https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html The Twin Paradox: The Equivalence Principle Analysis]&lt;/ref&gt;

&lt;ref name=cuvaj&gt;{{Citation |author=Cuvaj, C. |date=1971 |title=Paul Langevin and the theory of relativity|journal=Japanese studies in the history of science|volume=10| pages=113-142|url=http://www.isc.meiji.ac.jp/~sano/hssj/pdf/Cuvaj_C-1972-Langevin_Relativity-JSHS-No_10-pp113-142.pdf}}&lt;/ref&gt;

&lt;ref name=koks&gt;Koks, D. (2018): [https://math.ucr.edu/home/baez/physics/Relativity/SR/sr-gr.html Physics FAQ: Where is the Boundary between Special and General Relativity?]&lt;/ref&gt;

&lt;/references&gt;

[[Category:History of special relativity]]

[[Category:Paradoxes]]</text>
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    <title>Global Audiology/Asia/Philippines</title>
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{{CountryHeader|File:Philippines (orthographic projection).svg|https://en.wikipedia.org/wiki/Philippines}}{{HTitle|Brief Country Information }}
[https://en.wikipedia.org/wiki/Philippines The Philippines], officially the Republic of the Philippines, is an archipelagic country in Southeast Asia. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest. Filipino and English are the most spoken languages in the country, serving as the official languages, followed by Cebuano, Ilocano, Hiligaynon, Waray, Bikol and several other regional languages and dialects. 

{{HTitle|Incidence and Prevalence of Hearing Loss}}Hearing loss is a common problem across all age groups in the Philippines, and the prevalence rates are higher than global estimates (Newall et al., 2020). The national prevalence of moderate or worse hearing loss is 7.5% in children, 14.7% in working-age adults, and 49.1% in the elderly (Newall et al., 2020).

The prevalence rates are also high regionally, with a study conducted in Southern Tagalog Region IV-A: CALABARZON (Cavite, Laguna, Batangas, Rizal, and Quezon) showing that around 71% of people have at least mild hearing loss, and 26.33% have disabling hearing impairment (Pardo et al., 2022). Among children aged between 4 and 18 years, 11.87% have disabling hearing loss, while among working adults aged between 19 and 64 years, the prevalence rate is 8.97%. In older adults aged 65+, the rate is 3.17% (Pardo et al., 2022).

Local studies reveal that hereditary factors play a significant role. Research on cochlear implant patients found a frequent genetic mutation SLC26A4 c.706C&gt;G underlying hearing impairment (Chiong et al., 2018). Another study of an indigenous Filipino community identified genetic factors like SLC26A4 variants as major contributors to otitis media and conductive hearing loss (Santos-Cortez et al., 2016).

Outer and middle ear conditions, often caused by chronic infections, can also increase the risk of hearing impairment (Newall et al., 2020). Socioeconomic status is also a factor, with higher income associated with lower odds of moderate hearing loss. Wax occlusion affects 12.2% of people, while middle ear disease is present in 14.2% (Newall et al., 2020).

The number of Filipinos with severe to profound hearing loss is higher than in developed countries. Given the higher prevalence and severity rates, hearing loss is a significant public health concern that requires urgent attention to curb the rising disability.{{HTitle|Education and Professional Practice}}
Clinical audiology education is primarily offered at two institutions: the University of Santo Tomas (UST) and the University of the Philippines (UP). Both universities provide a Master's program in Clinical Audiology, which integrates theoretical knowledge with clinical applications. 

1. [https://www.ust.edu.ph/academics/programs/health/# UST Clinical Audiology Program]

The Master in Clinical Audiology at the University of Santo Tomas stands as a pioneering program in the Philippines, establishing itself as the nation's first two-year graduate degree in this specialized field. Since its inception over twenty years ago, the program has cultivated a network of nearly 300 clinical audiologists who now serve diverse healthcare settings both within the Philippines and internationally. 
Through strategic collaborations with leading universities, hospitals, and industry partners worldwide, the curriculum seamlessly integrates rigorous theoretical coursework with intensive hands-on clinical training. The program's commitment to excellence in audiology education, evidence-based practice, research innovation, and community outreach enables it to produce an average of 20 highly qualified clinical audiologists annually, contributing significantly to the advancement of hearing healthcare services.

2.[https://cm.upm.edu.ph/p/ms-clinical-audiology/ UP Clinical Audiology Program]

Jointly offered by the College of Medicine and the College of Allied Medical Professions, it exemplifies the university's commitment to academic excellence and community service. This collaborative initiative, supported by the Department of Otorhinolaryngology and the Department of Speech Pathology, aims to develop highly skilled audiologists who can address critical healthcare needs in hearing prevention, diagnosis, and treatment.

This two-year graduate program integrates comprehensive theoretical foundations with hands-on clinical experience. The curriculum encompasses four core areas: audiologic evaluation, audiologic habilitation, hearing conservation, and audiology service delivery program development. Through this rigorous training, graduates emerge prepared to deliver exceptional patient care while contributing to the advancement of audiological services and research.

The program's structure reflects the university's dual mission of providing outstanding advanced education and fostering meaningful community impact through professional healthcare services. By combining academic rigor with practical application, the program prepares the next generation of audiologists to meet evolving healthcare challenges and serve diverse community needs.{{HTitle|Challenges and Opportunities}}

=== Challenges ===
* '''Limited access to services'''
The prevalence of hearing loss in the Philippines is increasing, yet access to audiological services remains limited, particularly in rural and underserved areas. This lack of access highlights the need for expanded healthcare infrastructure and improved distribution of audiologists and trained hearing care healthcare workers across regions. Unfortunately, the disparities in the existing healthcare infrastructure significantly impact the quality of life of people living in these areas. 
* '''Shortage of qualified professionals''' 
The Philippines currently needs more skilled audiologists and other healthcare professionals specializing in hearing care. This shortage is impeding the nation's ability to meet the increasing demand for hearing healthcare services, highlighting the need for investment in education and training programs to develop a competent workforce.
* '''Brain drain'''
A significant challenge confronting audiology in the Philippines is the phenomenon of brain drain, wherein most locally-trained audiologists seek better opportunities overseas. This trend of emigration of skilled professionals exacerbates the shortfall of audiologists in the country and undermines initiatives to enhance the local healthcare system. The resulting shortage of qualified audiologists in the Philippines poses a severe concern for the delivery of ear and hearing care services and the population's overall well-being. Therefore, measures must be taken to address this issue and retain local talent in audiology.
* '''Lack of health insurance coverage for hearing devices'''
One significant problem in the Philippines is the need for health insurance coverage for hearing devices. This applies to both national insurance and private health insurance. As a result, individuals requiring hearing devices often have to pay for them out of their pockets or seek assistance from various sources. This includes seeking sponsorship from local politicians, NGOs, and social welfare services or receiving donations of refurbished hearing aids from other countries.
* '''Unregulated hearing centers and dispensing of hearing aids'''
The establishment of hearing centers and the distribution of hearing aids without appropriate licensing or training pose significant challenges to audiology practice in the Philippines. The lack of regulations and oversight results in substandard care, inaccurate diagnosis, and inappropriate management of hearing disorders, ultimately jeopardizing patient safety and outcomes. Implementing stringent guidelines and qualifications for hearing center establishments and hearing aid dispensers is imperative to ensure that qualified and trained professionals provide quality audiological care. Failure to do so may lead to detrimental consequences for patients with hearing impairments and the audiological profession.
*'''Awareness and stigma'''
In the Philippines, most of the general population lacks awareness and knowledge about hearing health issues. As a result of the stigma attached to hearing loss, people often delay seeking services, which can lead to insufficient management of auditory disorders. It is crucial to address this stigma through public education and awareness campaigns to promote early intervention, management, and (re)habilitation.

{{HTitle|References}}

# [https://psa.gov.ph/population-and-housing/node/186896Philippine Statistics Authority. (2022). 2020 Census of Population and Housing.]
# [https://www.ethnologue.com/country/PH Ethnologue. (2022). Languages of the World - Philippines.]
# Newall, J. P., Martinez, N., Swanepoel, W., &amp; McMahon, C. M. (2020). A National Survey of Hearing Loss in the Philippines. Asia-Pacific journal of public health, 32(5), 235–241. https://doi.org/10.1177/1010539520937086
# Pardo, P. M., Niñal-Vilog,  . , Acuin, J. M., Calaquian, C. E.,  &amp; Onofre-Telan, R. D.(2022).Hearing and clinical otologic profile of Filipinos living in Southern Tagalog Region IV-A (CALABARZON), Philippines: The Southern Tagalog ENT Hearing Specialists (STENTS) Survey 2012-2017. Philippine Journal of Otolaryngology Head and Neck Surgery, 37(2), 8-15
# Chiong, C. M., Reyes-Quintos, M. R. T., Yarza, T. K. L., Tobias-Grasso, C. A. M., Acharya, A., Leal, S. M., Mohlke, K. L., Mayol, N. L., Cutiongco-de la Paz, E. M., &amp; Santos-Cortez, R. L. P. (2018). The SLC26A4 c.706C&gt;G (p.Leu236Val) Variant is a Frequent Cause of Hearing Impairment in Filipino Cochlear Implantees. Otology &amp; neurotology : official publication of the American Otological Society, American Neurotology Society [and] European Academy of Otology and Neurotology, 39(8), e726–e730. https://doi.org/10.1097/MAO.0000000000001893
# Santos-Cortez, R. L., Reyes-Quintos, M. R., Tantoco, M. L., Abbe, I., Llanes, E. G., Ajami, N. J., Hutchinson, D. S., Petrosino, J. F., Padilla, C. D., Villarta, R. L., Jr, Gloria-Cruz, T. L., Chan, A. L., Cutiongco-de la Paz, E. M., Chiong, C. M., Leal, S. M., &amp; Abes, G. T. (2016). Genetic and Environmental Determinants of Otitis Media in an Indigenous Filipino Population. Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery, 155(5), 856–862. https://doi.org/10.1177/0194599816661703
# University of the Philippines Manila. (n.d.). Master in Clinical Audiology. College of Medicine. https://cm.upm.edu.ph/p/ms-clinical-audiology/
# University of Santo Tomas. (n.d.). Programs in health. https://www.ust.edu.ph/academics/programs/health/
# 
{{:Global Audiology/Authors-1|Joyce Rodvie Sagun|https://www.linkedin.com/in/joyce-rodvie-sagun-4691bb182}}
{{:Global Audiology/Footer}}

[[Category:Philippines]]
[[Category:Audiology]]</text>
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  <page>
    <title>Kidney</title>
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      <minor />
      <comment>First, fixed kidney location to the correct area, it is not located in sacral region. Second, added more precise explanation for reasons the right kidney is lower than the left. Previously stated &quot;the right kidney is pressed against the right kidney&quot;</comment>
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      <text bytes="3005" sha1="c8jlmypf60hhxa2r8tk2lgj2x6a6q1d" xml:space="preserve">Kidney (Latin ren, Greek. νεφρός [nephros], English. kidney) is a human organ. An adult has a length of 12 cm, a length of 6 cm, and a length of 3 cm. The kidney is located retroperitoneal of the abdominal organs, posterior to the abdominal wall&lt;ref&gt;{{Cite journal|date=2025-03-26|title=Kidney Anatomy: Overview, Gross Anatomy, Microscopic Anatomy|url=https://emedicine.medscape.com/article/1948775-overview?form=fpf}}&lt;/ref&gt;, at the 12th vertebra, and its minimum size is at the height of the 3rd dorsal vertebra. It is located 1.5-2.0 cm on the left and right sides. The right kidney is lower than the left one since it is displaced downwards by the liver &lt;ref&gt;{{Cite web|url=https://training.seer.cancer.gov/anatomy/urinary/components/kidney.html|title=Kidneys {{!}} SEER Training|website=training.seer.cancer.gov|access-date=2026-04-11}}&lt;/ref&gt;

== Kidney structure ==
The renal artery, the renal vein, and the ureter are all called the renal sinus. It is at the entrance of the kidney, and the cavity is called the renal pelvis. The renal pelvis collects the urine in the kidney and passes it to the ureter. The renal pelvis collects urine from 2-3 major calyces, and the major calyx collects urine from 2-3 minor calyces. The kidney is covered with fibrous connective tissue. Interstitial tissue of the kidney is a fluffy connective tissue, and on the outside it is rich in reticular and reticular economy. The kidney is divided into two parts: the cortex and the medulla, and the medulla, which is slightly more useful for the cortex, has acquired a pale appearance. The cerebrum is made up of 8-12 pyramidal structures (renal pyramids) with the base of the pyramid facing the cortex and the apex facing the entrance of the kidney. At the top of the pyramid, the tip of the papilla, such as the renal papilla, collects urine through a small cup. Interstitial tissue of the pyramids was deciphered as the renal column (renal column). The functional unit of the kidney is the nephron. Nephrons filter blood and produce urine, and each kidney consists of 1 million good nephrons.


[[File:Blausen 0592 KidneyAnatomy 01.png|thumb|structure of the human kidney]]

== Duty ==
Participates in the formation and excretion of urine.

Regulates cell fluid balance [ECF].

Regulates blood pressure.

Role of electrolyte compensation in body fluids.

Acid-base balance.

Regulates osmotic pressure in plasma.

Regulation of erythropoietin.

Renin-angiotensin, prostaglandin, and kallikrein-kinin apparatuses are involved in the endocrine system of the kidneys.

== How to protect your kidneys ==
The key to preventing and slowing down the progression of kidney disease is keeping your blood pressure stable, or more specifically, keeping your blood pressure below 130/80. 3/2 percent of the main causes of chronic kidney disease are diabetes and high blood pressure. That's why it is possible to prevent the disease by following a regular exercise and a healthy and well-balanced diet.

[[Category:Organs]]</text>
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  <page>
    <title>Bully Metric Rapinat</title>
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      <comment>Removing [[:c:File:Johannes_Kepler_1610.jpg|Johannes_Kepler_1610.jpg]], it has been deleted from Commons by [[:c:User:Aude|Aude]] because: File page with no file uploaded ([[:c:COM:CSD#F7|F7]]).</comment>
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|-
 | {{Original research}}
 | [https://physwiki.eeyabo.net/index.php/Main_Page &lt;small&gt;Development &lt;br/&gt;Area&lt;/small&gt;]
|}

[[Bully_Metric|Bully Metric Main Page]]&lt;br /&gt;
[[Bully_Metric_Rapinat|The Bully Metric rapidity unit]]&lt;br /&gt;
[[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]&lt;br /&gt;
[https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]&lt;br /&gt;

The '''rapinat''' (natural unit of [https://en.wikipedia.org/wiki/Rapidity rapidity]) (symbol '''Rn''') is defined such that an object with a [https://en.wikipedia.org/wiki/Standard_gravitational_parameter standard gravitational parameter] equal to the speed of light in vacuum cubed, multiplied by 30.55 femtoseconds, will have a gravitational mass of one rapinat timepan.  

 (mass = 1 Rn ta)
     ⇒ μ = [https://pml.nist.gov/cgi-bin/cuu/Value?c c&lt;sup&gt;3&lt;/sup&gt;] × 30.55 fs (exact)
     ⇒ μ ≈ [https://www.google.com/search?q=c%5E3+*+30.55+fs+in+km%5E3+/+s%5E2 823.139274 km^3 / s^2] (approximate)

Table 1 below was taken from the Wikipedia [https://en.wikipedia.org/wiki/Standard_gravitational_parameter standard gravitational parameter] article, and the mass of each body was calculated in Bully Metric transformation units:


{| class=&quot;wikitable&quot; style=&quot;margin:1em auto 1em auto; background:#fff; {{text color default}};&quot;
|+Table 1: The Bully Metric mass for selected solar system bodies 
|-
! Body
! colspan=&quot;2&quot;|'''''mass''''' [Rn ta]
|-
| [https://en.wikipedia.org/wiki/Sun Sun]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 161,227,199
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .646(12)
|-
| [https://en.wikipedia.org/wiki/Mercury_(planet) Mercury]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 26
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .765666(1)

|-
| [https://en.wikipedia.org/wiki/Venus Venus]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 394
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .658112(7)
 
|-
| [https://en.wikipedia.org/wiki/Earth Earth]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 484
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .244227(1)

|-
| [https://en.wikipedia.org/wiki/Mars Mars]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 52
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .03052(2)
|-
| [https://en.wikipedia.org/wiki/1_Ceres Ceres]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 0
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .076090
|-
| [https://en.wikipedia.org/wiki/Jupiter Jupiter]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 153,906
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .56(1)
|-
| [https://en.wikipedia.org/wiki/Saturn Saturn]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 46,081
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .13(1)
|-
| [https://en.wikipedia.org/wiki/Uranus Uranus]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 7,038
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .83(1)
|-
| [https://en.wikipedia.org/wiki/Neptune Neptune]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 8,305
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .43(1)
|-
| [https://en.wikipedia.org/wiki/Pluto Pluto]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 1
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .06(1)
|-
| [https://en.wikipedia.org/wiki/Eris_(dwarf_planet) Eris]
| style=&quot;text-align:right; border-right:none; padding-right:0;&quot; | 1
| style=&quot;text-align:left;  border-left: none; padding-left: 0;&quot; | .35(1)
|}

[[File:solar_system_mass_distribution_ppm_chart.svg|thumb|center|400px|[https://en.wikipedia.org/wiki/Parts-per_notation Parts-per-million] chart of the relative mass distribution of the Solar System, each cubelet denoting 161.4 Rn ta]]

== Gravitational mass ==
'''Active gravitational mass''' is a property of an object that produces a gravitational field in the space surrounding the object, and these gravitational fields govern large-scale structures in the [https://en.wikipedia.org/wiki/universe Universe].  Gravitational fields hold the [https://en.wikipedia.org/wiki/galaxies galaxies] together.  They cause clouds of gas and [https://en.wikipedia.org/wiki/dust dust] to coalesce into [https://en.wikipedia.org/wiki/stars stars] and [https://en.wikipedia.org/wiki/planets planets].  They provide the necessary pressure for [https://en.wikipedia.org/wiki/nuclear_fusion nuclear fusion] to occur within stars.  And they determine the [https://en.wikipedia.org/wiki/orbits orbits] of various objects within the [https://en.wikipedia.org/wiki/Solar_System Solar System].  Since gravitational effects are all around us, it is impossible to pin down the exact date when humans first discovered gravitational mass.  However, it is possible to identify some of the significant steps towards our modern understanding of gravitational mass and its relationship to the other mass phenomena. Some terms associated with gravitational mass and its effects are the [https://en.wikipedia.org/wiki/Gaussian_gravitational_constant Gaussian gravitational constant], the [https://en.wikipedia.org/wiki/standard_gravitational_parameter standard gravitational parameter] and the [https://en.wikipedia.org/wiki/Schwarzschild_radius Schwarzschild radius].

=== Keplerian gravitational mass ===

{|class=&quot;wikitable&quot; cellspacing=2 style=&quot;text-align:right&quot;
|-
!rowspan=2|English&lt;br&gt;name
!rowspan=8|
!colspan=3|The Keplerian planets
|- style=&quot;background:#ccc; {{text color default}};&quot;
![https://en.wikipedia.org/wiki/Semi-major_axis Semi-major axis]
![https://en.wikipedia.org/wiki/Sidereal_orbital_period Sidereal orbital period]
!Mass of Sun
|-
![https://en.wikipedia.org/wiki/Mercury_(planet) Mercury]
|0.387 099 [https://en.wikipedia.org/wiki/Astronomical_unit AU]
|0.240 842 [https://en.wikipedia.org/wiki/sidereal_year sidereal earth years]
|rowspan=6|&lt;math&gt;\propto 4\pi^2\frac{\text{AU}^3}{\text{y}^2}&lt;/math&gt;
|-
![https://en.wikipedia.org/wiki/Venus Venus]
|0.723 332 AU
|0.615 187 sidereal earth years
|-
![https://en.wikipedia.org/wiki/Earth Earth]
|1.000 000 AU
|1.000 000 sidereal earth years
|-
![https://en.wikipedia.org/wiki/Mars Mars]
|1.523 662 AU
|1.880 816 sidereal earth years
|-
![https://en.wikipedia.org/wiki/Jupiter Jupiter]
|5.203 363 AU
|11.861 776 sidereal earth years
|-
![https://en.wikipedia.org/wiki/Saturn Saturn]
|9.537 070 AU
|29.456 626 sidereal earth years
|}

[https://en.wikipedia.org/wiki/Johannes_Kepler Johannes Kepler] was the first to give an accurate description of the orbits of the planets, and by doing so; he was the first to describe gravitational mass.  In 1600 AD, Kepler sought employment with [https://en.wikipedia.org/wiki/Tycho_Brahe Tycho Brahe] and consequently gained access to astronomical data of a higher precision than any previously available.  Using Brahe’s precise observations of the planet Mars, Kepler realized that traditional astronomical methods were inaccurate in their predictions, and he spent the next five years developing his own method for characterizing planetary motion.

In Kepler’s final planetary model, he successfully described planetary orbits as following [https://en.wikipedia.org/wiki/elliptical elliptical] paths with the Sun at a focal point of the ellipse.  The concept of active gravitational mass is an immediate consequence of Kepler's [https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion third law of planetary motion].  Kepler discovered that the [https://en.wikipedia.org/wiki/square_(algebra) square] of the [https://en.wikipedia.org/wiki/orbital_period orbital period] of each planet is directly [https://en.wikipedia.org/wiki/Proportionality_(mathematics) proportional] to the [https://en.wikipedia.org/wiki/cube_(arithmetic) cube] of the [https://en.wikipedia.org/wiki/semi-major_axis semi-major axis] of its orbit, or equivalently, that the [https://en.wikipedia.org/wiki/ratio ratio] of these two values is constant for all planets in the [https://en.wikipedia.org/wiki/Solar_System Solar System].  This constant ratio is a direct measure of the Sun's active gravitational mass, it has units of distance cubed per time squared, and is known as the [https://en.wikipedia.org/wiki/standard_gravitational_parameter standard gravitational parameter]:

:&lt;math&gt;\mu=4\pi^2\frac{\text{distance}^3}{\text{time}^2}\propto\text{gravitational mass}&lt;/math&gt;


To convert to Bully Metric transformation units, we must divide the standard gravitational parameter of the Sun by c&lt;sup&gt;3&lt;/sup&gt; × 30.55 fs.

 [https://www.google.com/search?q=(4+*+pi^2+AU^3+%2F+year^2+)+%2F+(c^3+*+30.55+fs)&amp;oq=(4+*+pi^2+AU^3+%2F+year^2+)+%2F+(c^3+*+30.55+fs) &lt;math&gt;\left(\frac{4\pi^2\frac{\text{AU}^3}{\text{y}^2}}{30.55\ fs\ c^3}\right)\ Rn\ ta = 161\ 200\ 000\ Rn\ ta&lt;/math&gt; ]

==== Kepler's Three Laws in Bully Metric ====

[[Bully_Metric_Kepler_Laws|A Bully Metric formulation of Kepler's Three Laws]]

=== Galilean moons ===
[[File:Galileo.arp.300pix.jpg|left|100px|thumb| Galileo Galilei 1636.]]

{|class=&quot;wikitable&quot; cellspacing=2 style=&quot;text-align:right&quot;
|-
!rowspan=2|English&lt;br&gt;name !!rowspan=6| !!colspan=3|The Galilean moons
|- style=&quot;background:#ccc; {{text color default}};&quot;
![https://en.wikipedia.org/wiki/Semi-major_axis Semi-major axis]
![https://en.wikipedia.org/wiki/Sidereal_orbital_period Sidereal orbital period]
!Mass of Jupiter
|-
![https://en.wikipedia.org/wiki/Io_(moon) Io]
|0.002 819 AU
|0.004 843 sidereal earth years
|rowspan=5|&lt;math&gt;\propto 0.0038\ \pi^2\frac{\text{AU}^3}{\text{y}^2} &lt;/math&gt;
|-
![https://en.wikipedia.org/wiki/Europa_(moon) Europa]
|0.004 486 AU
|0.009 722 sidereal earth years
|-
![https://en.wikipedia.org/wiki/Ganymede_(moon) Ganymede]
|0.007 155 AU
|0.019 589 sidereal earth years
|-
![https://en.wikipedia.org/wiki/Callisto_(moon) Callisto]
|0.012 585 AU
|0.045 694 sidereal earth years
|}

In 1609, Johannes Kepler published his three rules known as Kepler's laws of planetary motion, explaining how the planets follow elliptical orbits under the influence of the Sun.  On 25 August of that same year, [https://en.wikipedia.org/wiki/Galileo_Galilei Galileo Galilei] demonstrated his first telescope to a group of Venetian merchants, and in early January of 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars.  However, after a few days of observation, Galileo realized that these &quot;stars&quot; were in fact orbiting Jupiter.  These four objects (later named the [https://en.wikipedia.org/wiki/Galilean_moons Galilean moons] in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun.  Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611 he had obtained remarkably accurate estimates for their periods.  Many years later, the semi-major axis of each moon was also estimated, thus allowing the gravitational mass of Jupiter to be determined from the orbits of its moons.  The gravitational mass of Jupiter was found to be approximately a thousandth of the gravitational mass of the Sun.

To convert to Bully Metric transformation units, we must divide the standard gravitational parameter of Jupiter by c&lt;sup&gt;3&lt;/sup&gt; × 30.55 fs.

 [https://www.google.com/search?q=(0.0038+*+pi^2+AU^3+%2F+year^2+)+%2F+(c^3+*+30.55+fs)&amp;oq=(0.0038+*+pi^2+AU^3+%2F+year^2+)+%2F+(c^3+*+30.55+fs) &lt;math&gt;\left(\frac{0.0038\pi^2\frac{\text{AU}^3}{\text{y}^2}}{30.55\ fs\ c^3}\right)\ Rn\ ta = 153\ 000\ Rn\ ta&lt;/math&gt; ]

=== Galilean free fall ===
[[File:Galileo.arp.300pix.jpg|left|100px|thumb| Galileo Galilei 1636.]]
[[File:Falling ball.jpg|thumb|right|150px|upright|Distance traveled by a freely falling ball is proportional to the square of the elapsed time.]]
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics.  However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,&lt;ref&gt;{{cite journal |last=Drake |first=S. |date=1979 |title=Galileo's Discovery of the Law of Free Fall |journal=Scientific American |volume=228 |issue=5 |pages=84–92 |bibcode=1973SciAm.228e..84D |doi=10.1038/scientificamerican0573-84}}&lt;/ref&gt; but the results obtained from these experiments were both realistic and compelling.  A biography by Galileo's pupil [https://en.wikipedia.org/wiki/Vincenzo_Viviani Vincenzo Viviani] stated that Galileo had dropped [https://en.wikipedia.org/wiki/Ball ball]s of the same material, but different masses, from the [https://en.wikipedia.org/wiki/Leaning_Tower_of_Pisa Leaning Tower of Pisa] to demonstrate that their time of descent was independent of their mass.&lt;ref group=&quot;note&quot;&gt;At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version that predicted that bodies ''of the same material'' falling through the same medium would fall at the same speed. See {{cite book |last=Drake |first=S. |date=1978 |title=Galileo at Work |pages=[https://archive.org/details/galileoatwork00stil/page/19 19–20] |publisher=University of Chicago Press |isbn=978-0-226-16226-3 |url=https://archive.org/details/galileoatwork00stil/page/19 }}&lt;/ref&gt; In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?  The only convincing resolution to this question is that all bodies must fall at the same rate.&lt;ref&gt;{{cite book |last=Galileo |first=G. |date=1632 |title=Dialogue Concerning the Two Chief World Systems|title-link=Dialogue Concerning the Two Chief World Systems }}&lt;/ref&gt;

A later experiment was described in Galileo's ''Two New Sciences'' published in 1638.  One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp.  The wooden ramp was &quot;12 cubits long, half a cubit wide and three finger-breadths thick&quot; with a straight, smooth, polished [https://en.wikipedia.org/wiki/Groove_(engineering) Groove (engineering)|groove].  The groove was lined with &quot;[https://en.wikipedia.org/wiki/Parchment parchment], also smooth and polished as possible&quot;.  And into this groove was placed &quot;a hard, smooth and very round bronze ball&quot;.  The ramp was inclined at various [https://en.wikipedia.org/wiki/Angle angle]s to slow the acceleration enough so that the elapsed time could be measured.  The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured.  The time was measured using a water clock described as follows:
:a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.&lt;ref&gt;{{cite book |last1=Galileo |first1=G. |date=1638 |title=Discorsi e Dimostrazioni Matematiche, Intorno à Due Nuove Scienze |volume=213 |publisher=[[House of Elzevir|Louis Elsevier]]}}, translated in {{cite book |date=1954 |editor1-last=Crew |editor1-first=H. |editor2-last=de Salvio |editor2-first=A. |title=Mathematical Discourses and Demonstrations, Relating to Two New Sciences |url=http://oll.libertyfund.org/index.php?option=com_staticxt&amp;staticfile=show.php%3Ftitle=753&amp;Itemid=99999999 |publisher=[[Dover Publications]] |isbn=978-1-275-10057-2 |access-date=11 April 2012 |archive-date=1 October 2013 |archive-url=https://web.archive.org/web/20131001015122/http://oll.libertyfund.org/index.php?option=com_staticxt&amp;staticfile=show.php%3Ftitle=753&amp;Itemid=99999999 |url-status=dead }} and also available in {{cite book |editor1-last=Hawking |editor1-first=S. |date=2002 |title=On the Shoulders of Giants |pages=[https://archive.org/details/isbn_9780762413485/page/534 534–535] |publisher=[[Running Press]] |isbn=978-0-7624-1348-5 |url=https://archive.org/details/isbn_9780762413485/page/534 }}&lt;/ref&gt;

Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:
: &lt;math&gt;{\text{Distance}} \propto {\text{Time}^2}&lt;/math&gt;

Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass.  However, Galileo's free fall motions and Kepler's planetary motions remained distinct concepts during Galileo's lifetime.

== Newtonian mass ==
[[File:GodfreyKneller-IsaacNewton-1689.jpg|left|100px|thumb| Isaac Newton, 1689.]]

{|class=&quot;wikitable&quot; cellspacing=2 style=&quot;text-align:right&quot;
! colspan=2| Earth's Moon !! rowspan=2| Mass of Earth
|-
![https://en.wikipedia.org/wiki/Semi-major_axis Semi-major axis]
![https://en.wikipedia.org/wiki/Sidereal_orbital_period Sidereal orbital period]
|-
|0.002 569 AU||0.074 802 sidereal years
|rowspan=3|&lt;math qid=Q11376&gt;1.2\pi^2\cdot10^{-5}\frac{\text{AU}^3}{\text{y}^2}=3.986\cdot10^{14}\frac{\text{m}^3}{\text{s}^2}&lt;/math&gt;
|-
! Earth's gravity !! Earth's radius
|-
|9.806 65&amp;nbsp;m/s&lt;sup&gt;2&lt;/sup&gt;||6 375&amp;nbsp;km
|}

[https://en.wikipedia.org/wiki/Robert_Hooke Robert Hooke] published his concept of gravitational forces in 1674, stating that all [https://en.wikipedia.org/wiki/Astronomical_object celestial bodies] have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center.&lt;ref&gt;{{cite book |last=Hooke |first=R. |date=1674 |title=An attempt to prove the motion of the earth from observations |url=https://books.google.com/books?id=JgtPAAAAcAAJ&amp;pg=PA1 |publisher=[[Royal Society]]}}&lt;/ref&gt; In correspondence with [https://en.wikipedia.org/wiki/Isaac_Newton Isaac Newton] from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the square of the distance between the two bodies.&lt;ref&gt;{{cite book |editor-last=Turnbull |editor-first=H.W. |date=1960 |title=Correspondence of Isaac Newton, Volume 2 (1676–1687) |page=297 |publisher=Cambridge University Press}}&lt;/ref&gt; Hooke urged Newton, who was a pioneer in the development of [https://en.wikipedia.org/wiki/Calculus calculus], to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct.  Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke.  Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend, [https://en.wikipedia.org/wiki/Edmond_Halley Edmond Halley], that he had solved the problem of gravitational orbits, but had misplaced the solution in his office.&lt;ref&gt;{{cite book |title=Principia |url=https://massless.info/images/Isaac_Newton_Principia_English.pdf |pages=16}}&lt;/ref&gt; After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings.  In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled ''[https://en.wikipedia.org/wiki/De_motu_corporum_in_gyrum De motu corporum in gyrum]'' (Latin for &quot;On the motion of bodies in an orbit&quot;).&lt;ref&gt;
{{cite book |editor-last=Whiteside |editor-first=D.T. |date=2008 |title=The Mathematical Papers of Isaac Newton, Volume VI (1684–1691) |url=https://books.google.com/books?id=lIZ0v23iqRgC|publisher=Cambridge University Press |isbn=978-0-521-04585-8}}&lt;/ref&gt; Halley presented Newton's findings to the [https://en.wikipedia.org/wiki/Royal_Society Royal Society] of London, with a promise that a fuller presentation would follow.  Newton later recorded his ideas in a three-book set, entitled ''[https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica Philosophiæ Naturalis Principia Mathematica]'' (English: ''Mathematical Principles of Natural Philosophy'').  The first was received by the Royal Society on 28 April 1685–86; the second on 2 March 1686–87; and the third on 6 April 1686–87.  The Royal Society published Newton's entire collection at their own expense in May 1686–87.&lt;ref name=&quot;principia&quot;&gt;{{cite book |author1=Sir Isaac Newton |author2=N.W. Chittenden |title=Newton's Principia: The mathematical principles of natural philosophy |url=https://archive.org/details/newtonsprincipi00chitgoog |page=[https://archive.org/details/newtonsprincipi00chitgoog/page/n43 31]|year=1848 |publisher=D. Adee|isbn=9780520009295 }}&lt;/ref&gt;{{rp|31}}

Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these:
: &lt;math&gt;\mathbf{g}=-\mu\frac{\hat{\mathbf{R}}}{|\mathbf{R}|^2}&lt;/math&gt;
where '''g''' is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, ''μ'' is the gravitational mass ([https://en.wikipedia.org/wiki/Standard_gravitational_parameter standard gravitational parameter]) of the body causing gravitational fields, and '''R''' is the radial coordinate (the distance between the centers of the two bodies).

By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass.  The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius.  The mass of the Earth is approximately three-millionths of the mass of the Sun.  To date, no other accurate method for measuring gravitational mass has been discovered.&lt;ref&gt;{{cite web |last=Cuk |first=M. |date=January 2003 |title=Curious About Astronomy: How do you measure a planet's mass? |url=https://curious.astro.cornell.edu/question.php?number=452 |work=Ask an Astronomer |access-date=2011-03-12 |url-status=dead |archive-url=https://web.archive.org/web/20030320113723/http://curious.astro.cornell.edu/question.php?number=452 |archive-date=20 March 2003}}&lt;/ref&gt;

To convert to Bully Metric transformation units, we must divide the standard gravitational parameter of the Earth by c&lt;sup&gt;3&lt;/sup&gt; × 30.55 fs.

 [https://www.google.com/search?q=(0.000012+*+pi^2+AU^3+%2F+year^2+)+%2F+(c^3+*+30.55+fs)&amp;oq=(0.000012+*+pi^2+AU^3+%2F+year^2+)+%2F+(c^3+*+30.55+fs) &lt;math&gt;\left(\frac{0.000012\pi^2\frac{\text{AU}^3}{\text{y}^2}}{30.55\ fs\ c^3}\right)\ Rn\ ta = 484\ Rn\ ta&lt;/math&gt; ]

=== Newton's cannonball ===
[[File:Newton Cannon.svg|thumb|A cannon on top of a very high mountain shoots a cannonball horizontally.  If the speed is low, the cannonball quickly falls back to Earth (A,&amp;nbsp;B). At [https://en.wikipedia.org/wiki/Orbital_speed intermediate speeds], it will revolve around Earth along an elliptical orbit (C,&amp;nbsp;D). Beyond the [https://en.wikipedia.org/wiki/Escape_velocity escape velocity], it will leave the Earth without returning (E).]]
[https://en.wikipedia.org/wiki/Newton%27s_cannonball Newton's cannonball]

Newton's cannonball was a [https://en.wikipedia.org/wiki/Thought_experiment thought experiment] used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits.  It appeared in Newton's 1728 book ''A Treatise of the System of the World''.  According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth.  However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path.  &quot;For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground.  And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth.&quot;&lt;ref name=principia /&gt;{{rp|513}} Newton further reasons that if an object were &quot;projected in an horizontal direction from the top of a high mountain&quot; with sufficient velocity, &quot;it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected.&quot;&lt;ref&gt;{{cite book |last1=Newton |first1=Isaac |author-link=Isaac Newton |title=A Treatise of the System of the World |date=1728 |publisher=F. Fayram |location=London |page=6 |url=https://books.google.com/books?id=rEYUAAAAQAAJ&amp;q=ball&amp;pg=PR1 |access-date=4 May 2022}}&lt;/ref&gt;

== Gravitational Mass as an Amount ==
===Pre-Newtonian concepts===
{{main|weight}}
[[File:El pesado del corazón en el Papiro de Hunefer.jpg|right|thumb|Depiction of early [[balance scales]] in the [[Papyrus of Hunefer]] (dated to the [[Nineteenth dynasty of Egypt|19th dynasty]], {{circa|1285 BCE}}). The scene shows [[Anubis]] weighing the heart of Hunefer.]]
The concept of [[wikt:amount|amount]] is very old and [[Prehistoric numerals|predates recorded history]].  Humans, at some early era, realized that the weight of a collection of similar objects was [[Proportionality (mathematics)|directly proportional]] to the number of objects in the collection:

: &lt;math&gt;W_n \propto n,&lt;/math&gt;

where ''W'' is the weight of the collection of similar objects and ''n'' is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant [[ratio]]:

: &lt;math&gt;\frac{W_n}{n} = \frac{W_m}{m}&lt;/math&gt;, or equivalently &lt;math&gt;\frac{W_n}{W_m} = \frac{n}{m}.&lt;/math&gt;

An early use of this relationship is a [[balance scale]], which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields.  Hence, if they have similar masses then their weights will also be similar.  This allows the scale, by comparing weights, to also compare masses.

Consequently, historical weight standards were often defined in terms of amounts.  The Romans, for example, used the [[carob]] seed ([[Carat (unit)|carat]] or [[siliqua]]) as a measurement standard.  If an object's weight was equivalent to [http://std.dkuug.dk/JTC1/SC2/WG2/docs/n3138.pdf 1728 carob seeds], then the object was said to weigh one Roman pound.  If, on the other hand, the object's weight was equivalent to [[Ancient Roman units of measurement|144 carob seeds]] then the object was said to weigh one Roman ounce (uncia).  The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed.  The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was:

: &lt;math&gt;\frac{\mathrm{ounce}}{\mathrm{pound}} = \frac{W_{144}}{W_{1728}} = \frac{144}{1728} = \frac{1}{12}.&lt;/math&gt;

===Earth's Gravitational Mass Amount===
To put things in perspective, the modern British stone has a mass of slightly less than 19.4 Roman pounds (approximately 33 514 carob seeds), and the Earth has a mass of slightly less than 10&lt;sup&gt;24&lt;/sup&gt; British stones.  To make things precise, let the &quot;Bully Stone&quot; be defined to have a gravitational mass of exactly 500 Rn yta, and with that definition in mind, the mass of the Earth can be calculated as:

* Mass of Earth = 968 488 455 000 000 000 000 000 Bully Stones.
* 1 Bully Stone := 500 Rn yta (approximately [https://www.google.com/search?q=500+*+10%5E%28-24%29+*+30.55+fs+*+c%5E3+%2F++G 6.1665 kilograms]).
* Mass of one Bully Stone ≈ 32 544 carob seeds.

===Newton's Universal gravitational mass===
[[File:Universal gravitational mass.PNG|An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center|left|thumb]]
Robert Hooke asserted in 1674 that: &quot;all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers&quot;, but Hooke had neither explained why this gravitating attraction was unique to celestial bodies, nor had he explained why the attraction was directed towards the center of a celestial body. To answer these questions, Newton introduced the entirely new concept that gravitational mass is &quot;universal&quot;: meaning that every object has gravitational mass, and therefore, every object generates a gravitational field.  Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object.  With these assumptions in mind, Newton calculated what the overall gravitational field would be if a large collection of small objects were formed into a giant spherical body.  Newton found that a giant spherical body (like the Earth or Sun, with roughly uniform density at each given radius), would have a gravitational field which was proportional to the total mass of the body,&lt;ref name=principia /&gt;{{rp|397}} and inversely proportional to the square of the distance to the body's center.&lt;ref name=principia /&gt;{{rp|221}}

As explained previously, the modern British stone has a mass equivalent to approximately 33 514 carob seeds, and the Earth has a mass of slightly less than 10&lt;sup&gt;24&lt;/sup&gt; British stones. According to Newton's theory of universal gravitation, each carob seed produces gravitational fields.  Therefore, in principle, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere.  Hence, it is theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun.

===The Cavendish Experiment===
[[File:Cavendish Experiment.png|thumb|right|Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.]]

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice.  According to Newton's theory all objects produce gravitational fields, however, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure.  And if one were to collect an immense number of objects, the resulting sphere would probably be too large to construct on the surface of the Earth, and too expensive to construct in space.  Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the [[Cavendish experiment]], did not occur until 1797, over a hundred years later.  Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water.  As of 2009, the Earth's mass in &quot;kilograms&quot; is only known to around five digits of accuracy, whereas its gravitational mass in &quot;Bully Stones&quot; is given above to nine significant figures.

 Mass of Earth =   968 488 455 000 000 000 000 000 Bully Stones.
 Mass of Earth = 5 972 200 000 000 000 000 000 000 kilograms.

== Gravitational Fields ==

[[File:Spherical_shell_with_gravity.svg|An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center|left|thumb]]
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== Notes ==
{{NoteFoot}}

== References ==
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[[File:Sorbus torminalis Trunk and canopy.jpg|thumb|300px|The canopy of a Checker tree &lt;small&gt;(''Torminalis glaberrima'')&lt;/small&gt;]]&lt;br /&gt;
 
Most of my wiki contributions are made to [[:species:Main Page|Wikispecies]] where I'm an administrator, bureaucrat and interface admin,&lt;small&gt;&lt;sup&gt;[https://species.wikimedia.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist (verify)]&lt;/sup&gt;&lt;/small&gt; to the Swedish Wikimedia Chapter [[WMSE:|Wikimedia Sverige]] (WMSE) where I'm an administrator,&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://se.wikimedia.org/w/index.php?title=Special:Användare&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt; and as administrator and interface administrator at the Swedish version of [[wikivoyage:sv:Huvudsida|Wikivoyage]].&lt;small&gt;&lt;sup&gt;(&lt;span class=&quot;plainlinks&quot;&gt;[https://sv.wikivoyage.org/w/index.php?title=Special:ListUsers&amp;limit=1&amp;username=Tommy_Kronkvist verify]&lt;/span&gt;)&lt;/sup&gt;&lt;/small&gt;

So far (April 12, 2026), I've made just over 391,100 edits to 153 of the Wikimedia sister projects&amp;nbsp;– the majority of them to Wikispecies and Wikidata. My global account information for all of Wikimedia can be found [[meta:Special:CentralAuth/Tommy Kronkvist|here]].

Swedish is my mother tongue&amp;nbsp;– even though I was born in Finland&amp;nbsp;– but I feel comfortable speaking and writing English and to some extent in German as well. Odd as it may seem, unfortunately I can't speak any Finnish even though I went to school there for a few years prior to moving to Sweden (see [[w:Swedish-speaking population of Finland|Swedish-speaking population of Finland]] in Wikipedia). I've lived all over Sweden but nowadays reside in Uppsala, the fourth biggest city and former capital of Sweden.

I'm only the fourth generation named &quot;Kronkvist&quot;. My family name consists of two parts: ''kron'' – a short form of the Swedish word ''krona'' meaning 'crown', as in coronation crown or tree crown – and ''kvist'', meaning 'bough' or 'twig'. Hence the name ''Kronkvist'' refers to a twig in the canopy of a forest. I'm the fourth generation of Kronkvist's. Prior to that our family name was ''Mattus'': an oeconym meaning &quot;Matthew's Farm&quot;, dating back to at least 1637.
{{Clear}}
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[[File:ChatGPT-Logo.svg|thumb|Was ChatGPT's essay on Kohlberg's theory of moral development accurate? Should we use AI in our own workfields? ]]
''The following human-created article '''assesses''' an AI-generated essay, and this page, therefore, does not contain AI-generated content. The ChatGPT-generated article is linked [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing/ChatGPT essay|here]].''

Aaqib F. Azeez, January 2026

'''Abstract:''' The usage of artificial intelligence has raised significant questions regarding its accuracy and reliability. This paper assesses an essay created by ChatGPT (Model 5.2) on Lawrence Kohlberg's Cognitive-Developmental Theory regarding moral development and modern viewpoints on the theory. The AI-generated essay demonstrated several strengths, including accurately describing Kohlberg's stages and using authentic and academically-appropriate sources. Despite the positives, the essay also had significant drawbacks, including a lack of citations for strong claims. The findings in the paper point to several positive contributions AI can have in academic and professional tasks, but this should be treated with caution and preferably overseen by competent professionals.

''Keywords:'' artificial intelligence, ChatGPT, academia, workforce, Lawrence Kohlberg's stages of moral development, citation accuracy{{Italic title}}
{{Tertiary education}}
{{psychology}}
{{paper}}
{{complete}}
== Introduction ==
The usage of AI has caused a lot of chaos in the academic world. One of the issues with the usage of AI is the accuracy of its work. Although AI is convenient for brainstorming ideas or providing a framework to work and improve on, using AI for a finished product is counterintuitive, since AI can hallucinate details to please the user. The purpose of this paper is to assess the accuracy of AI by evaluating an essay made by the [[w:ChatGPT|ChatGPT]] model 5.2 on [[w:Lawrence_Kohlberg's_stages_of_moral_development|Lawrence Kohlberg’s Cognitive-Developmental Theory]]. 

== Critical analysis ==
The essay created by ChatGPT hosts a variety of positives. Firstly, the in-text citations correctly correspond to the references listed at the bottom of the page. For example, “Killen &amp; Dahl, 2021” correctly corresponds to the reference “Killen, M., &amp; Dahl, A. (2021)” listed under “References”. Secondly, all the publications ChatGPT used were published later than 2018, according to the instructions given to the AI model. Thirdly, the in-text citations were accurate to the sources. For example, the paper cites Mammen &amp; Paulus (2023) to support the assertion that moral reasoning should be examined under natural conversations with other people, and not just a structured interview&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mammen|first=Maria|last2=Paulus|first2=Markus|date=2023-04-01|title=The communicative nature of moral development: A theoretical framework on the emergence of moral reasoning in social interactions|url=https://www.sciencedirect.com/science/article/pii/S0885201423000412|journal=Cognitive Development|volume=66|pages=101336|doi=10.1016/j.cogdev.2023.101336|issn=0885-2014}}&lt;/ref&gt;. Mammen &amp; Paulus (2023) support the AI’s statements since the original paper notes that interviews can only study the final product of decision-making, therefore missing perceptions on the process of decision-making&lt;ref name=&quot;:0&quot; /&gt;. Mammen &amp; Paulus (2023) upheld the importance of moral development and its origin “rooted in human communication.”&lt;ref name=&quot;:0&quot; /&gt; The essay provided by the AI is also accurate, correctly identifying each stage of Kohlberg’s Cognitive-Developmental Theory. The essay mentions that the preconventional level is characterized by moral judgment that focuses on “punishment avoidance or personal benefit”, the conventional level is a witness to the beginning of “social approval, rules, and [the maintenance of] social order” influencing morality, and the postconventional stage being the final level where morality ascends to the “broader principles such as justice, rights, and fairness that may sometimes conflict with authority”. Santrock (2025) describes Kohlberg’s first level in a similar way, painting the preconventional stage as being “strongly influenced by external punishment and reward”, describing the conventional reasoning stage as the site of “understand[ing] the importance of following the laws of society”, and reports the postconventional reasoning stage as the “highest” level, where norms are being pitted against “moral concerns such as liberty, justice, and equality, with the idea that morality can improve the laws” (p. 228)&lt;ref&gt;Santrock, J. W. (2025). ''Adolescence'' (19th ed.). McGraw Hill Higher Education&lt;/ref&gt;. Lastly, the AI-generated essay stays on topic and doesn’t deviate from the main subject and provides the readers with up-to-date information on the subject. The essay discusses the basics of Kohlberg’s cognitive-developmental theory (including the 3 stages) and includes modern developments of Kohlberg’s theories, including the integration of individual differences, culture, social contexts, and emotions. 

Although the essay is largely accurate and useful, it is marred by a few shortcomings. The AI-generated essay makes the claim that a “common critique” of Kohlberg’s theory is that Kohlberg’s description of “advanced reasoning” is not always practical in “real-world moral decisions, especially under stress or social pressure.” Since the essay claims that this is a “common critique”, a citation to back this up should be provided. Another instance of this is where ChatGPT claims that the stage theories have received criticism for “assumptions about universality and the primacy of justice-based reasoning.” The same issue is repeated here, where there are no citations or sources provided to justify this claim. The lack of sources for broad claims that the AI has made does not decimate the credibility of the essay but hampers the ability to verify and fact-check certain statements that hold modern weight.

== Usage of AI in the real world ==
The addition of AI in professional settings can compensate for deficiencies in the speed at which humans collect information. Notable positive traits of AI include its speedy collection and examination of data, lol ability to improve with minimal human aid after training, and the ability to carry out crucial decision-making processes based on analytical reasoning&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=Jarrahi|first=Mohammad Hossein|date=2018-07-01|title=Artificial intelligence and the future of work: Human-AI symbiosis in organizational decision making|url=https://www.sciencedirect.com/science/article/pii/S0007681318300387|journal=Business Horizons|volume=61|issue=4|pages=577–586|doi=10.1016/j.bushor.2018.03.007|issn=0007-6813}}&lt;/ref&gt;. A 2021 Deloitte and MedTech Europe report projected that AI could potentially save 380,000-403,000 lives per year in European healthcare&lt;ref&gt;Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., &amp; Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. &lt;nowiki&gt;https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next&lt;/nowiki&gt;&lt;/ref&gt;. Even from the AI-generated essay, AI can not only provide a well-written description of an entity but also include accurate citations that correspond with the text. The combination of AI’s analytical pace and the human’s heightened judgement and integration of “[[w:Abstract_thinking|abstract thinking]]” and “intuitive approach” could bolster each other’s performance in certain tasks, as observed in a 2016 study reported by Jarrahi (2018) that saw an 85% reduction in error in cancer detection when AI and pathologists both collaborated on the task&lt;ref name=&quot;:1&quot; /&gt;. With AI’s ability to process and analyze information efficiently and quickly, it can prove to be a handy tool that can bolster human production.

Despite AI’s efficient processing of information, AI has been notorious for hallucinating information, as was observed in a 2023 incident where a group of lawyers from [https://www.lawinfo.com/lawfirm/new-york/new-york/levidow-levidow-and-oberman-pc/6d64c18e-3681-4881-a187-1a3628604b27.html Levidow, Levidow &amp; Oberman, P.C.] were caught publishing fabricated court decisions marred with “fake quotes and citations created by the artificial intelligence tool ChatGPT”&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://apnews.com/article/artificial-intelligence-chatgpt-fake-case-lawyers-d6ae9fa79d0542db9e1455397aef381c|title=Lawyers submitted bogus case law created by ChatGPT. A judge fined them $5,000|date=2023-06-22|website=AP News|language=en|access-date=2026-01-02}}&lt;/ref&gt;. Other issues include a lack of security for confidential data, a lack of accountability for when harm may be produced by AI, and the increased likelihood of “engaging in unethical behavior” when using AI&lt;ref&gt;{{Cite journal|last=Trincado-Munoz|first=Francisco J.|last2=Cordasco|first2=Carlo|last3=Vorley|first3=Tim|date=2025-04-26|title=The dark side of AI in professional services|url=https://doi.org/10.1080/02642069.2024.2336208|journal=The Service Industries Journal|volume=45|issue=5-6|pages=455–474|doi=10.1080/02642069.2024.2336208|issn=0264-2069}}&lt;/ref&gt;. Although the AI-generated essay was largely positive and the citations were accurate, there were certain claims that were not backed up by a source. As mentioned earlier, the essay claimed that Kohlberg’s theories were criticized for their lack of practicality in real-world scenarios yet provided no citation to back this claim. It is worthwhile that when engaging with AI, it can be used for “assistance”, but oversight of such work should be done to “ensure [its] accuracy”&lt;ref name=&quot;:2&quot; /&gt;.

== Conclusion ==
The AI-generated essay was, for the most part, exceptional. The essay stayed on topic, provided accurate developments to theory with respect to the modern era, and contained accurate citations and references, but failed to back up a couple of strong claims. AI has shown itself to be quick in collecting and processing information, but supervision over its work should be done to ensure accuracy. AI should be introduced into professional settings as it clearly is a useful tool, but the notion of AI technology ‘replacing’ humans in the workforce is unfounded and would lead to more harm than commendable.

== See also ==

* [https://zenodo.org/records/18135581 Azeez, A. (2026). ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing. Zenodo. https://doi.org/10.5281/zenodo.18135581]- APA-compliant paper
* [https://www.academia.edu/145733465/ChatGPTs_Essay_on_Kohlbergs_Theory_AIs_Use_in_Academic_Writing Academia link]
* [https://www.theihs.org/blog/best-practices-for-using-ai-in-academic-research/?gad_source=1&amp;gad_campaignid=22511278759&amp;gbraid=0AAAAADkVWeGyeqRyzexNn7a9sGkUIKevh&amp;gclid=Cj0KCQiA9t3KBhCQARIsAJOcR7zTf2F5CkrUJZAHLM-myxLhFkzx4dWRpNpQE2UIB4u1werlmLZMAAMaAlybEALw_wcB Best Practices for Using AI in Academic Research - Institute for Humane Studies]

== References ==
{{reflist}}
[[Category:Atcovi's Work]]
[[Category:ChatGPT]]
[[Category:Essays]]
[[Category:Developmental psychology]]</text>
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[[File:ChatGPT-Logo.svg|thumb|Was ChatGPT's essay on Kohlberg's theory of moral development accurate? Should we use AI in our own workfields? ]]
''The following human-created article '''assesses''' an AI-generated essay, and this page, therefore, does not contain AI-generated content. The ChatGPT-generated article is linked [[ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing/ChatGPT essay|here]].''

Aaqib F. Azeez, January 2026

'''Abstract:''' The usage of artificial intelligence has raised significant questions regarding its accuracy and reliability. This paper assesses an essay created by ChatGPT (Model 5.2) on Lawrence Kohlberg's Cognitive-Developmental Theory regarding moral development and modern viewpoints on the theory. The AI-generated essay demonstrated several strengths, including accurately describing Kohlberg's stages and using authentic and academically-appropriate sources. Despite the positives, the essay also had significant drawbacks, including a lack of citations for strong claims. The findings in the paper point to several positive contributions AI can have in academic and professional tasks, but this should be treated with caution and preferably overseen by competent professionals.

''Keywords:'' artificial intelligence, ChatGPT, academia, workforce, Lawrence Kohlberg's stages of moral development, citation accuracy{{Italic title}}
{{Tertiary education}}
{{psychology}}
{{paper}}
{{complete}}
== Introduction ==
The usage of AI has caused a lot of chaos in the academic world. One of the issues with the usage of AI is the accuracy of its work. Although AI is convenient for brainstorming ideas or providing a framework to work and improve on, using AI for a finished product is counterintuitive, since AI can hallucinate details to please the user. The purpose of this paper is to assess the accuracy of AI by evaluating an essay made by the [[w:ChatGPT|ChatGPT]] model 5.2 on [[w:Lawrence_Kohlberg's_stages_of_moral_development|Lawrence Kohlberg’s Cognitive-Developmental Theory]]. 

== Critical analysis ==
The essay created by ChatGPT hosts a variety of positives. Firstly, the in-text citations correctly correspond to the references listed at the bottom of the page. For example, “Killen &amp; Dahl, 2021” correctly corresponds to the reference “Killen, M., &amp; Dahl, A. (2021)” listed under “References”. Secondly, all the publications ChatGPT used were published later than 2018, according to the instructions given to the AI model. Thirdly, the in-text citations were accurate to the sources. For example, the paper cites Mammen &amp; Paulus (2023) to support the assertion that moral reasoning should be examined under natural conversations with other people, and not just a structured interview&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=Mammen|first=Maria|last2=Paulus|first2=Markus|date=2023-04-01|title=The communicative nature of moral development: A theoretical framework on the emergence of moral reasoning in social interactions|url=https://www.sciencedirect.com/science/article/pii/S0885201423000412|journal=Cognitive Development|volume=66|pages=101336|doi=10.1016/j.cogdev.2023.101336|issn=0885-2014}}&lt;/ref&gt;. Mammen &amp; Paulus (2023) support the AI’s statements since the original paper notes that interviews can only study the final product of decision-making, therefore missing perceptions on the process of decision-making&lt;ref name=&quot;:0&quot; /&gt;. Mammen &amp; Paulus (2023) upheld the importance of moral development and its origin “rooted in human communication.”&lt;ref name=&quot;:0&quot; /&gt; The essay provided by the AI is also accurate, correctly identifying each stage of Kohlberg’s Cognitive-Developmental Theory. The essay mentions that the preconventional level is characterized by moral judgment that focuses on “punishment avoidance or personal benefit”, the conventional level is a witness to the beginning of “social approval, rules, and [the maintenance of] social order” influencing morality, and the postconventional stage being the final level where morality ascends to the “broader principles such as justice, rights, and fairness that may sometimes conflict with authority”. Santrock (2025) describes Kohlberg’s first level in a similar way, painting the preconventional stage as being “strongly influenced by external punishment and reward”, describing the conventional reasoning stage as the site of “understand[ing] the importance of following the laws of society”, and reports the postconventional reasoning stage as the “highest” level, where norms are being pitted against “moral concerns such as liberty, justice, and equality, with the idea that morality can improve the laws” (p. 228)&lt;ref&gt;Santrock, J. W. (2025). ''Adolescence'' (19th ed.). McGraw Hill Higher Education&lt;/ref&gt;. Lastly, the AI-generated essay stays on topic and doesn’t deviate from the main subject and provides the readers with up-to-date information on the subject. The essay discusses the basics of Kohlberg’s cognitive-developmental theory (including the 3 stages) and includes modern developments of Kohlberg’s theories, including the integration of individual differences, culture, social contexts, and emotions. 

Although the essay is largely accurate and useful, it is marred by a few shortcomings. The AI-generated essay makes the claim that a “common critique” of Kohlberg’s theory is that Kohlberg’s description of “advanced reasoning” is not always practical in “real-world moral decisions, especially under stress or social pressure.” Since the essay claims that this is a “common critique”, a citation to back this up should be provided. Another instance of this is where ChatGPT claims that the stage theories have received criticism for “assumptions about universality and the primacy of justice-based reasoning.” The same issue is repeated here, where there are no citations or sources provided to justify this claim. The lack of sources for broad claims that the AI has made does not decimate the credibility of the essay but hampers the ability to verify and fact-check certain statements that hold modern weight.

== Usage of AI in the real world ==
The addition of AI in professional settings can compensate for deficiencies in the speed at which humans collect information. Notable positive traits of AI include its speedy collection and examination of data, the ability to improve with minimal human aid after training, and the ability to carry out crucial decision-making processes based on analytical reasoning&lt;ref name=&quot;:1&quot;&gt;{{Cite journal|last=Jarrahi|first=Mohammad Hossein|date=2018-07-01|title=Artificial intelligence and the future of work: Human-AI symbiosis in organizational decision making|url=https://www.sciencedirect.com/science/article/pii/S0007681318300387|journal=Business Horizons|volume=61|issue=4|pages=577–586|doi=10.1016/j.bushor.2018.03.007|issn=0007-6813}}&lt;/ref&gt;. A 2021 Deloitte and MedTech Europe report projected that AI could potentially save 380,000-403,000 lives per year in European healthcare&lt;ref&gt;Dantas, C., Mackiewicz, K., Tageo, V., Jacucci, G., Guardado, D., Ortet, S., Varlamis, I., Maniadakis, M., De Lera, E., Quintas, J., Kocsis, O., &amp; Vassiliou, C. (2021). Benefits and hurdles of AI in the workplace – what comes next? ''International Journal of Artificial Intelligence and Expert Systems, 10'', 9-17. &lt;nowiki&gt;https://www.researchgate.net/publication/351993615_Benefits_and_Hurdles_of_AI_In_The_Workplace_-What_Comes_Next&lt;/nowiki&gt;&lt;/ref&gt;. Even from the AI-generated essay, AI can not only provide a well-written description of an entity but also include accurate citations that correspond with the text. The combination of AI’s analytical pace and the human’s heightened judgement and integration of “[[w:Abstract_thinking|abstract thinking]]” and “intuitive approach” could bolster each other’s performance in certain tasks, as observed in a 2016 study reported by Jarrahi (2018) that saw an 85% reduction in error in cancer detection when AI and pathologists both collaborated on the task&lt;ref name=&quot;:1&quot; /&gt;. With AI’s ability to process and analyze information efficiently and quickly, it can prove to be a handy tool that can bolster human production.

Despite AI’s efficient processing of information, AI has been notorious for hallucinating information, as was observed in a 2023 incident where a group of lawyers from [https://www.lawinfo.com/lawfirm/new-york/new-york/levidow-levidow-and-oberman-pc/6d64c18e-3681-4881-a187-1a3628604b27.html Levidow, Levidow &amp; Oberman, P.C.] were caught publishing fabricated court decisions marred with “fake quotes and citations created by the artificial intelligence tool ChatGPT”&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://apnews.com/article/artificial-intelligence-chatgpt-fake-case-lawyers-d6ae9fa79d0542db9e1455397aef381c|title=Lawyers submitted bogus case law created by ChatGPT. A judge fined them $5,000|date=2023-06-22|website=AP News|language=en|access-date=2026-01-02}}&lt;/ref&gt;. Other issues include a lack of security for confidential data, a lack of accountability for when harm may be produced by AI, and the increased likelihood of “engaging in unethical behavior” when using AI&lt;ref&gt;{{Cite journal|last=Trincado-Munoz|first=Francisco J.|last2=Cordasco|first2=Carlo|last3=Vorley|first3=Tim|date=2025-04-26|title=The dark side of AI in professional services|url=https://doi.org/10.1080/02642069.2024.2336208|journal=The Service Industries Journal|volume=45|issue=5-6|pages=455–474|doi=10.1080/02642069.2024.2336208|issn=0264-2069}}&lt;/ref&gt;. Although the AI-generated essay was largely positive and the citations were accurate, there were certain claims that were not backed up by a source. As mentioned earlier, the essay claimed that Kohlberg’s theories were criticized for their lack of practicality in real-world scenarios yet provided no citation to back this claim. It is worthwhile that when engaging with AI, it can be used for “assistance”, but oversight of such work should be done to “ensure [its] accuracy”&lt;ref name=&quot;:2&quot; /&gt;.

== Conclusion ==
The AI-generated essay was, for the most part, exceptional. The essay stayed on topic, provided accurate developments to theory with respect to the modern era, and contained accurate citations and references, but failed to back up a couple of strong claims. AI has shown itself to be quick in collecting and processing information, but supervision over its work should be done to ensure accuracy. AI should be introduced into professional settings as it clearly is a useful tool, but the notion of AI technology ‘replacing’ humans in the workforce is unfounded and would lead to more harm than commendable.

== See also ==

* [https://zenodo.org/records/18135581 Azeez, A. (2026). ChatGPT's Essay on Kohlberg's Theory: AI's Use in Academic Writing. Zenodo. https://doi.org/10.5281/zenodo.18135581]- APA-compliant paper
* [https://www.academia.edu/145733465/ChatGPTs_Essay_on_Kohlbergs_Theory_AIs_Use_in_Academic_Writing Academia link]
* [https://www.theihs.org/blog/best-practices-for-using-ai-in-academic-research/?gad_source=1&amp;gad_campaignid=22511278759&amp;gbraid=0AAAAADkVWeGyeqRyzexNn7a9sGkUIKevh&amp;gclid=Cj0KCQiA9t3KBhCQARIsAJOcR7zTf2F5CkrUJZAHLM-myxLhFkzx4dWRpNpQE2UIB4u1werlmLZMAAMaAlybEALw_wcB Best Practices for Using AI in Academic Research - Institute for Humane Studies]

== References ==
{{reflist}}
[[Category:Atcovi's Work]]
[[Category:ChatGPT]]
[[Category:Essays]]
[[Category:Developmental psychology]]</text>
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= ChrisP =

* '''My contributions:''' [[Special:Contributions/ChrisP]]

== My pages ==
{{Special:PrefixIndex/DesignWriteStudio/Course/StudentPages/ChrisP/|hideredirects=1|stripprefix=1}}

== My assignment checklist ==
* 1.1 → [[DesignWriteStudio/Course/StudentPages/ChrisP/1.1 Defining Hypertext]]
* 1.2 → [[DesignWriteStudio/Course/StudentPages/ChrisP/1.2 Hypertext in the Wild]]
* 2.1 → [[DesignWriteStudio/Course/StudentPages/ChrisP/2.1 Hypertext Feature Encyclopedia Entry]]
* 2.2 → [[DesignWriteStudio/Course/StudentPages/ChrisP/2.2 Expanded Hypertext Feature Encyclopedia Entry]]
* 2.3 → [[DesignWriteStudio/Course/StudentPages/ChrisP/2.3 Validating an encyclopedia entry written by another author]]
* 3.1 → [[DesignWriteStudio/Course/StudentPages/ChrisP/3.1 Hypertext Examples Fiction Gaming]]
* 3.2 → [[DesignWriteStudio/Course/StudentPages/ChrisP/3.2 Hypertext Examples Catalogues Datastores]]
* 3.3 → [[DesignWriteStudio/Course/StudentPages/ChrisP/3.3 Hypertext Design Challenges Introduction]]
* 3.4 → [[DesignWriteStudio/Course/StudentPages/ChrisP/3.4 Hypertext Design Challenges Advanced]]
https://en.wikiversity.org/wiki/DesignWriteStudio/Course/StudentPages/ChrisP/4.2_Portfolio

{{:DesignWriteStudio/SiteElements/Footer}}
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&lt;div style=&quot;text-align: center&quot;&gt;

[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]&lt;br&gt;
Friday 24 April, 2026 11:00 - 12:00 AEST

[https://utsmeet.zoom.us/j/84179400467 Zoom link]

&lt;!-- Slides TBA (Google)&lt;br&gt;
Video TBA (YouTube; 53:37 mins including Q&amp;A)
[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post]
[https://twitter.com/jtneill/status/1768516693553565884 (example) X post]
--&gt;
&lt;/div&gt;
==Overview==
{{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}}
{{RoundBoxTop|theme=3}}
Many student assignments are written for one person, read once, and then never read again. 

In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. 

Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio.

The session will explore: 
* What open wiki assignments look like in practice, and where they go wrong
* The realities of working in publicly editable spaces (including having work changed or deleted)
* Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around

This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching.
{{RoundBoxBottom}}

==Introduction==
[[w:Wiki|Wiki]]s were developed in the early days of the world wide web (1994) as the simplest web page that anyone can edit (see [[w:History of wikis|history of wikis]]. Today, wikis are the foundation of the remarkable global open knowledge projects developed by the Wikimedia Foundation. These interconnected wiki projects, such as Wikipedia, are tuned to the lofty goal of making the sum of all human knowledge freely available to all—and editable by anyone. These pages are commonly visited, openly licensed, often updated, and are used to train artificial intelligence.

[[File:Wiki project case study onion diagram.svg|right|thumb|'''Figure 1'''. An onion layer model of open wiki assignments for authentic learning]]
Wikis are also grand social experiments. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and communicate with the public (see Figure 1). Wikis can be used to develop disciplinary knowledge, interact in a dynamic social learning and editing environment, and to foster generic skills and graduate attributes such as communication skills and being able to make creative use of technology.

Staff and students contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. The work is immediately available to the public and can be edited by anyone. This radical transparency can be challenging for staff and students but ultimately empowers capacity and confidence in contributing directly to the knowledge commons.  

Wiki-based assignment formats are flexible and can vary widely depending on subject area, level of study, and targetted skills, but often involve contributing, curating, and improving text or media (images, audio, and video) which can be presented as open educational resources, encyclopedic articles, books, articles, manuals, journals, structured data sets, and so on.

Open educational wikis can function as [[w:Content management system|content management systems]] for hosting open teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. While wikis can support the development of open textbooks, they also enable more diverse, collaborative, and participatory forms of knowledge production than institutionally supported textbook platforms such as [[w:PressBooks|PressBooks]]. In the context of [[w:Tertiary education in Australia|Australian higher education]], such platforms are typically staff-controlled, with limited opportunities for student authorship and co-creation.

Wikis give ongoing access to learning materials beyond student graduation, and staff have access regardless of  institutional employment. Concerns about content curation are resolved by discussion and consensus. Materials can also be forked, like software, to allow different development directions.

==Wikimedia projects==
===Wikipedia===
The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists.

The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. 

However, I would cast the net wider than Wikipedia because:
* Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence.
* Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources.

For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects.

===Beyond Wikipedia===

Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment.

{{sisterprojects}}
Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development.

An open wiki higher education assignment generally involves:
* Students contributing discipline-relevant content to the global knowledge commons via one or more of the Wikimedia Foundation sister projects 
* Assignment tasks centre on producing and refining useful knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences
* Outputs can include text, media, data, and learning resources  
* Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages])  

Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources.

{| class=&quot;wikitable&quot; style=&quot;margin: 0 auto;&quot;
|+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments
! Project
! Purpose
! Example assignments
|-
| [[b:|Wikibooks]]
| New books (e.g., textbooks)
| 
* Contribute to development of an open textbook
* Curate and improve existing OER book chapters
* Package a series of related articles into a new book
|-
| [[commons:|Wikimedia Commons]]
| Images, audio, and video
|
* Contribute high-quality educational media
* Improve metadata and categorisation
* Create educational diagrams and illustrations
* Upload field recordings or interviews
|-
| [[d:Wikidata|Wikidata]]
| Structured, linked open data
|
* Create and curate datasets
* Link concepts across Wikimedia projects
* Model relationships between entities
* Support data-driven research and analysis
|-
| [[q:|Wikiquote]]
| Quotations
|
* Curate and improve text quotes from primary sources such as political speeches
* Create categories for quotes by theme or topic
* Add citations and verification to existing quotes
|-
| [[species:|Wikispecies]]
| Taxonomy and species classification
|
* Curate and improve taxonomic entries for species
* Add citations for classification and nomenclature
* Contribute information about newly described species
* Improve links between species and related Wikimedia projects
|-
| [[s:Main Page|Wikisource]]
| Primary texts and historical documents
|
* Transcribe and proofread source texts
* Annotate and contextualise historical documents
* Curate thematic collections of primary sources
|-
| [[v:|Wikiversity]]
| Learning, teaching, and research
|
* Create open educational resources
* Develop teaching materials (e.g., lesson plans, self-assessment quizzes)
* Publish student research project summaries
* Improve existing learning resources by adding new text and multimedia
|-
| [[voy:Main Page|Wikivoyage]]
| Travel guides and geographic knowledge
|
* Develop place-based guides (e.g., regions, cities)
* Contribute cultural, historical, or environmental information
* Integrate fieldwork or experiential learning outputs
|-
| [[wikt:Main Page|Wiktionary]]
| Lexical and linguistic resources
|
* Create and refine dictionary entries
* Analyse word meanings, usage, and etymology
* Contribute multilingual translations and examples
|-
| [[w:|Wikipedia]]
| Encyclopedic information
|
* Contribute to articles related to the class topic where a gap exists
* Improve the quality and accuracy of existing articles
* Add citations and references to unverified text
* Curate and improve a category of articles related to a specific subject area
|}

==Open wiki assignments==
Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available.

Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include:
* '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s).
* '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators.
* '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions.
* '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]].
*'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. 
* '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research.

Advantages of open wiki assignments include:
* '''Perpetuity''' - ongoing availability of resources
* '''Linkability''' - cross-linking of projects and external resources
* '''Editability''' - resources can be improved by anyone
* '''Discussability''' - each resource has a discussion page
* '''Showability''' - resources showcase curator skills and knowledge
* '''Transparency''' - resource edit history and can be reviewed
* '''Forkability''' - open licence allows development of alternative resources

==Examples==
Here are some examples of open wiki assignments:
* [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia 
* [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia
* [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA

Whilst not assignments per se, these innovative open wiki resources may inspire:
* [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries

==Activities==
* Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]]
* Edit your [[Help:User page|Wikiversity user page]]
* Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]]
* Brainstorm what you or your students could contribute
* Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support

==Bio==
[[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''.

==See also==
;Wikimedia Foundation
* [[meta:Education|Education]] (Global WMF hub)
* [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia)

;Wikipedia
* [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta)

;Wikiversity
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article)
* [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page)
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation)

[[Category:User:Jtneill/Presentations/Open education]]
[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
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&lt;div style=&quot;text-align: center&quot;&gt;

[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]&lt;br&gt;
Friday 24 April, 2026 11:00 - 12:00 AEST

[https://utsmeet.zoom.us/j/84179400467 Zoom link]

&lt;!-- Slides TBA (Google)&lt;br&gt;
Video TBA (YouTube; 53:37 mins including Q&amp;A)
[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post]
[https://twitter.com/jtneill/status/1768516693553565884 (example) X post]
--&gt;
&lt;/div&gt;
==Overview==
{{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}}
{{RoundBoxTop|theme=3}}
Many student assignments are written for one person, read once, and then never read again. 

In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. 

Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio.

The session will explore: 
* What open wiki assignments look like in practice, and where they go wrong
* The realities of working in publicly editable spaces (including having work changed or deleted)
* Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around

This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching.
{{RoundBoxBottom}}

==Introduction==
[[w:Wiki|Wiki]]s were developed in the early days of the world wide web (1994) as the simplest web page that anyone can edit (for more, see [[w:History of wikis|history of wikis]]). Today, wikis are the foundation for the global open knowledge projects developed by the Wikimedia Foundation, the best known of which is Wikipedia. These dozen or so sister wiki projects have in common the lofty goal of making the sum of all human knowledge freely available to all—and allowing that knowledge to be editable by anyone. These projects have a high impact because they are quickly listed by search engines, often visited, updatable, used to train artificial intelligence, and openly licensed to allow use by others.

[[File:Wiki project case study onion diagram.svg|right|thumb|'''Figure 1'''. An onion layer model of open wiki assignments for authentic learning]]
Wiki technology enables grand social experiments. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and communicate openly with the public (see Figure 1). Wiki projects can be used to develop disciplinary knowledge and social and communication skills by making creative use of the Wikimedia Foundation wiki projects as freely available, active, open knowledge, collaborative online technology platforms.

Participants (staff, students, others) contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. The work is immediately available to the public and can be edited by anyone. In this way the open educational resources are iteratively improved. The radical transparency of open editing can intially be challenging for university staff and students but ultimately this approach empowers participants' capacity and confidence in contributing directly to the knowledge commons and optimises the utility of the work.

Wiki-based assignment formats can vary depending on:
* Subject area (e.g., humanities, sciences, professional fields)
* Level of study (e.g., informal, K-12, undergraduate, postgraduate, professional learning)
* Intended learning outcomes and development of generic skills (e.g., critical thinking, communication, digital literacy)

Typical tasks involve creating, curating, or improving open educational resources such as:
* Books (e.g., open textbooks) — Wikibooks
* Data (e.g., linked data items and data analysis) — Wikidata
* Conferences (e.g., home page, applications, abstracts, program, presentations) — Wikiversity
* Encyclopedic articles — Wikipedia
* Essays and analytical reports — Wikiversity
* Images, audio, and video (with metadata and licensing) — Wikimedia Commons
* Fact sheets and study guides — Wikiversity
* Journal-style articles and literature reviews — Wikiversity
* Learning activities (e.g., lectures, tutorials, workshops) — Wikiversity
* Manuals, tutorials, and how-to guides — Wikiversity

Open educational wikis can serve as [[w:Content management system|content management systems]] for hosting teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. Wikis can support the development of open textbooks. Compared to popular institutionally supported textbook platforms for open textbooks such as [[w:PressBooks|PressBooks]], there are no fees to use Wikimedia projects and they enable more diverse, collaborative, and participatory knowledge production.&lt;ref group=&quot;note&quot;&gt;In the context of [[w:Tertiary education in Australia|Australian higher education]], such platforms such as PressBooks are typically staff-controlled, with limited opportunities for student authorship and co-creation.&lt;/ref&gt;

A notable feature of wikis is that every page has a complete, searchable edit history. Each revision can be reviewed and, if necessary, reverted, ensuring that no content is permanently lost. Most edits incrementally improve the quality of a page; however, a small proportion are unhelpful and are therefore undone. As a rough guide, approximately 95% of edits are retained, while around 5% are reverted or deleted.

Wikis support long-term knowledge preservation. For students, an open wiki assignment ensures that their work remains publicly available and that they retain access to course materials  materials beyond graduation. For staff, access to open wiki materials continues regardless of institutional affiliation.

On wikis, disagreements about content are addressed through open discussion and consensus-building. This creates a distinctive collaborative environment in which students develop core skills in argumentation, communication, and negotiation. Wiki content can also be readily forked, similar to open-source software, enabling alternative versions to evolve in parallel. There is also a need for translation and development of open knowledge materials in different languages. As a result, wikis foster a pragmatic, solution-focused culture for collaborative knowledge development.

==Wikimedia projects==
===Wikipedia===
The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists.

The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. 

However, I would cast the net wider than Wikipedia because:
* Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence.
* Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources.

For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects.

===Beyond Wikipedia===

Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment.

{{sisterprojects}}
Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development.

An open wiki higher education assignment generally involves:
* Students contributing discipline-relevant content to the global knowledge commons via one or more of the Wikimedia Foundation sister projects 
* Assignment tasks centre on producing and refining useful knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences
* Outputs can include text, media, data, and learning resources  
* Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages])  

Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources.

{| class=&quot;wikitable&quot; style=&quot;margin: 0 auto;&quot;
|+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments
! Project
! Purpose
! Example assignments
|-
| [[b:|Wikibooks]]
| New books (e.g., textbooks)
| 
* Contribute to development of an open textbook
* Curate and improve existing OER book chapters
* Package a series of related articles into a new book
|-
| [[commons:|Wikimedia Commons]]
| Images, audio, and video
|
* Contribute high-quality educational media
* Improve metadata and categorisation
* Create educational diagrams and illustrations
* Upload field recordings or interviews
|-
| [[d:Wikidata|Wikidata]]
| Structured, linked open data
|
* Create and curate datasets
* Link concepts across Wikimedia projects
* Model relationships between entities
* Support data-driven research and analysis
|-
| [[q:|Wikiquote]]
| Quotations
|
* Curate and improve text quotes from primary sources such as political speeches
* Create categories for quotes by theme or topic
* Add citations and verification to existing quotes
|-
| [[species:|Wikispecies]]
| Taxonomy and species classification
|
* Curate and improve taxonomic entries for species
* Add citations for classification and nomenclature
* Contribute information about newly described species
* Improve links between species and related Wikimedia projects
|-
| [[s:Main Page|Wikisource]]
| Primary texts and historical documents
|
* Transcribe and proofread source texts
* Annotate and contextualise historical documents
* Curate thematic collections of primary sources
|-
| [[v:|Wikiversity]]
| Learning, teaching, and research
|
* Create open educational resources
* Develop teaching materials (e.g., lesson plans, self-assessment quizzes)
* Publish student research project summaries
* Improve existing learning resources by adding new text and multimedia
|-
| [[voy:Main Page|Wikivoyage]]
| Travel guides and geographic knowledge
|
* Develop place-based guides (e.g., regions, cities)
* Contribute cultural, historical, or environmental information
* Integrate fieldwork or experiential learning outputs
|-
| [[wikt:Main Page|Wiktionary]]
| Lexical and linguistic resources
|
* Create and refine dictionary entries
* Analyse word meanings, usage, and etymology
* Contribute multilingual translations and examples
|-
| [[w:|Wikipedia]]
| Encyclopedic information
|
* Contribute to articles related to the class topic where a gap exists
* Improve the quality and accuracy of existing articles
* Add citations and references to unverified text
* Curate and improve a category of articles related to a specific subject area
|}

==Open wiki assignments==
Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available.

Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include:
* '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s).
* '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators.
* '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions.
* '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]].
*'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. 
* '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research.

Advantages of open wiki assignments include:
* '''Perpetuity''' - ongoing availability of resources
* '''Linkability''' - cross-linking of projects and external resources
* '''Editability''' - resources can be improved by anyone
* '''Discussability''' - each resource has a discussion page
* '''Showability''' - resources showcase curator skills and knowledge
* '''Transparency''' - resource edit history and can be reviewed
* '''Forkability''' - open licence allows development of alternative resources

==Examples==
Here are some examples of open wiki assignments:
* [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia 
* [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia
* [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA

Whilst not assignments per se, these innovative open wiki resources may inspire:
* [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries

==Activities==
* Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]]
* Edit your [[Help:User page|Wikiversity user page]]
* Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]]
* Brainstorm what you or your students could contribute
* Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support

==Bio==
[[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''.

==See also==
;Wikimedia Foundation
* [[meta:Education|Education]] (Global WMF hub)
* [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia)

;Wikipedia
* [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta)

;Wikiversity
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article)
* [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page)
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation)

[[Category:User:Jtneill/Presentations/Open education]]
[[Category:User:Jtneill/Presentations/Wikiversity]]</text>
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&lt;div style=&quot;text-align: center&quot;&gt;

[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]&lt;br&gt;
Friday 24 April, 2026 11:00 - 12:00 AEST

[https://utsmeet.zoom.us/j/84179400467 Zoom link]

&lt;!-- Slides TBA (Google)&lt;br&gt;
Video TBA (YouTube; 53:37 mins including Q&amp;A)
[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post]
[https://twitter.com/jtneill/status/1768516693553565884 (example) X post]
--&gt;
&lt;/div&gt;
==Overview==
{{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}}
{{RoundBoxTop|theme=3}}
Many student assignments are written for one person, read once, and then never read again. 

In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. 

Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio.

The session will explore: 
* What open wiki assignments look like in practice, and where they go wrong
* The realities of working in publicly editable spaces (including having work changed or deleted)
* Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around

This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching.
{{RoundBoxBottom}}

==Introduction==
[[w:Wiki|Wiki]]s were developed in the early days of the world wide web (1994) as the simplest web page that anyone can edit (for more, see [[w:History of wikis|history of wikis]]). Today, wikis are the foundation for the global open knowledge projects developed by the Wikimedia Foundation, the best known of which is Wikipedia. These dozen or so sister wiki projects have in common the lofty goal of making the sum of all human knowledge freely available to all—and allowing that knowledge to be editable by anyone. These projects have a high impact because they are quickly listed by search engines, often visited, updatable, used to train artificial intelligence, and openly licensed to allow use by others.

[[File:Wiki project case study onion diagram.svg|right|thumb|'''Figure 1'''. An onion layer model of open wiki assignments for authentic learning]]
Wiki technology enables grand social experiments. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and communicate openly with the public (see Figure 1). Wiki projects can be used to develop disciplinary knowledge and social and communication skills by making creative use of the Wikimedia Foundation wiki projects as freely available, active, open knowledge, collaborative online technology platforms.

Participants (staff, students, others) contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. The work is immediately available to the public and can be edited by anyone. In this way the open educational resources are iteratively improved. The radical transparency of open editing can intially be challenging for university staff and students but ultimately this approach empowers participants' capacity and confidence in contributing directly to the knowledge commons and optimises the utility of the work.

Wiki-based assignment formats can vary depending on:
* Subject area (e.g., humanities, sciences, professional fields)
* Level of study (e.g., informal, K-12, undergraduate, postgraduate, professional learning)
* Intended learning outcomes and development of generic skills (e.g., critical thinking, communication, digital literacy)

Typical tasks involve creating, curating, or improving open educational resources such as:
* Books (e.g., open textbooks) — Wikibooks
* Data (e.g., linked data items and data analysis) — Wikidata
* Conferences (e.g., home page, applications, abstracts, program, presentations) — Wikiversity
* Encyclopedic articles — Wikipedia
* Essays and analytical reports — Wikiversity
* Images, audio, and video (with metadata and licensing) — Wikimedia Commons
* Fact sheets and study guides — Wikiversity
* Journal-style articles and literature reviews — Wikiversity
* Learning activities (e.g., lectures, tutorials, workshops) — Wikiversity
* Manuals, tutorials, and how-to guides — Wikiversity

Open educational wikis can serve as [[w:Content management system|content management systems]] for hosting teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. Wikis can support the development of open textbooks. Compared to popular institutionally supported textbook platforms for open textbooks such as [[w:PressBooks|PressBooks]], there are no fees to use Wikimedia projects and they enable more diverse, collaborative, and participatory knowledge production.&lt;ref group=&quot;note&quot;&gt;In the context of [[w:Tertiary education in Australia|Australian higher education]], such platforms such as PressBooks are typically staff-controlled, with limited opportunities for student authorship and co-creation.&lt;/ref&gt;

A notable feature of wikis is that every page has a complete, searchable edit history. Each revision can be reviewed and, if necessary, reverted, ensuring that no content is permanently lost. Most edits incrementally improve the quality of a page; however, a small proportion are unhelpful and are therefore undone. As a rough guide, approximately 95% of edits are retained, while around 5% are reverted or deleted.

Wikis support long-term knowledge preservation. For students, an open wiki assignment ensures that their work remains publicly available and that they retain access to course materials  materials beyond graduation. For staff, access to open wiki materials continues regardless of institutional affiliation.

On wikis, disagreements about content are addressed through open discussion and consensus-building. This creates a distinctive collaborative environment in which students develop core skills in argumentation, communication, and negotiation. Wiki content can also be readily forked, similar to open-source software, enabling alternative versions to evolve in parallel. There is also a need for translation and development of open knowledge materials in different languages. As a result, wikis foster a pragmatic, solution-focused culture for collaborative knowledge development.

==Wikimedia projects==
===Wikipedia===
The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists.

The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. 

However, I would cast the net wider than Wikipedia because:
* Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence.
* Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources.

For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects.

===Beyond Wikipedia===

Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment.

{{sisterprojects}}
Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development.

An open wiki higher education assignment generally involves:
* Students contributing discipline-relevant content to the global knowledge commons via one or more of the Wikimedia Foundation sister projects 
* Assignment tasks centre on producing and refining useful knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences
* Outputs can include text, media, data, and learning resources  
* Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages])  

Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources.

{| class=&quot;wikitable&quot; style=&quot;margin: 0 auto;&quot;
|+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments
! Project
! Purpose
! Example assignments
|-
| [[b:|Wikibooks]]
| New books (e.g., textbooks)
| 
* Contribute to development of an open textbook
* Curate and improve existing OER book chapters
* Package a series of related articles into a new book
|-
| [[commons:|Wikimedia Commons]]
| Images, audio, and video
|
* Contribute high-quality educational media
* Improve metadata and categorisation
* Create educational diagrams and illustrations
* Upload field recordings or interviews
|-
| [[d:Wikidata|Wikidata]]
| Structured, linked open data
|
* Create and curate datasets
* Link concepts across Wikimedia projects
* Model relationships between entities
* Support data-driven research and analysis
|-
| [[q:|Wikiquote]]
| Quotations
|
* Curate and improve text quotes from primary sources such as political speeches
* Create categories for quotes by theme or topic
* Add citations and verification to existing quotes
|-
| [[species:|Wikispecies]]
| Taxonomy and species classification
|
* Curate and improve taxonomic entries for species
* Add citations for classification and nomenclature
* Contribute information about newly described species
* Improve links between species and related Wikimedia projects
|-
| [[s:Main Page|Wikisource]]
| Primary texts and historical documents
|
* Transcribe and proofread source texts
* Annotate and contextualise historical documents
* Curate thematic collections of primary sources
|-
| [[v:|Wikiversity]]
| Learning, teaching, and research
|
* Create open educational resources
* Develop teaching materials (e.g., lesson plans, self-assessment quizzes)
* Publish student research project summaries
* Improve existing learning resources by adding new text and multimedia
|-
| [[voy:Main Page|Wikivoyage]]
| Travel guides and geographic knowledge
|
* Develop place-based guides (e.g., regions, cities)
* Contribute cultural, historical, or environmental information
* Integrate fieldwork or experiential learning outputs
|-
| [[wikt:Main Page|Wiktionary]]
| Lexical and linguistic resources
|
* Create and refine dictionary entries
* Analyse word meanings, usage, and etymology
* Contribute multilingual translations and examples
|-
| [[w:|Wikipedia]]
| Encyclopedic information
|
* Contribute to articles related to the class topic where a gap exists
* Improve the quality and accuracy of existing articles
* Add citations and references to unverified text
* Curate and improve a category of articles related to a specific subject area
|}

==Open wiki assignments==
Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available.

Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include:
* '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s).
* '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators.
* '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions.
* '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]].
*'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. 
* '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research.

Advantages of open wiki assignments include:
* '''Perpetuity''' - ongoing availability of resources
* '''Linkability''' - cross-linking of projects and external resources
* '''Editability''' - resources can be improved by anyone
* '''Discussability''' - each resource has a discussion page
* '''Showability''' - resources showcase curator skills and knowledge
* '''Transparency''' - resource edit history and can be reviewed
* '''Forkability''' - open licence allows development of alternative resources

==Examples==
Here are some examples of open wiki assignments:
* [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia 
* [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia
* [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA

Whilst not assignments per se, these innovative open wiki resources may inspire:
* [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries

==Activities==
* Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]]
* Edit your [[Help:User page|Wikiversity user page]]
* Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]]
* Brainstorm what you or your students could contribute
* Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support

==Bio==
[[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''.

==See also==
;Wikimedia Foundation
* [[meta:Education|Education]] (Global WMF hub)
* [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia)

;Wikipedia
* [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta)

;Wikiversity
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article)
* [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page)
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation)

[[Category:User:Jtneill/Presentations/Open education]]
[[Category:User:Jtneill/Presentations/Wikiversity]]

== Notes ==
&lt;references group=&quot;note&quot; /&gt;&lt;!--
== References ==
&lt;references /&gt;
--&gt;</text>
      <sha1>96y5lwda48pv41xn49wb5vf5dhfvuoe</sha1>
    </revision>
    <revision>
      <id>2804465</id>
      <parentid>2804464</parentid>
      <timestamp>2026-04-12T11:06:11Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Introduction */</comment>
      <origin>2804465</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18699" sha1="9627slg4v57j56yrhxuc4jwl7uz41ra" xml:space="preserve">{{title|Open wiki assignments for authentic learning}}
&lt;div style=&quot;text-align: center&quot;&gt;

[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]&lt;br&gt;
Friday 24 April, 2026 11:00 - 12:00 AEST

[https://utsmeet.zoom.us/j/84179400467 Zoom link]

&lt;!-- Slides TBA (Google)&lt;br&gt;
Video TBA (YouTube; 53:37 mins including Q&amp;A)
[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post]
[https://twitter.com/jtneill/status/1768516693553565884 (example) X post]
--&gt;
&lt;/div&gt;
==Overview==
{{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}}
{{RoundBoxTop|theme=3}}
Many student assignments are written for one person, read once, and then never read again. 

In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. 

Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio.

The session will explore: 
* What open wiki assignments look like in practice, and where they go wrong
* The realities of working in publicly editable spaces (including having work changed or deleted)
* Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around

This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching.
{{RoundBoxBottom}}

==Introduction==
[[w:Wiki|Wiki]]s were developed in the early days of the world wide web (1994) as the simplest web page that anyone can edit (for more, see [[w:History of wikis|history of wikis]]). Today, wikis are the foundation for the global open knowledge projects developed by the Wikimedia Foundation, the best known of which is Wikipedia. These dozen or so sister wiki projects have in common the lofty goal of making the sum of all human knowledge freely available to all—and allowing that knowledge to be editable by anyone. These projects have a high impact because they are quickly listed by search engines, often visited, updatable, used to train artificial intelligence, and openly licensed to allow use by others.

[[File:Wiki project case study onion diagram.svg|right|thumb|'''Figure 1'''. An onion layer model of open wiki assignments for authentic learning]]
Wiki technology enables grand social experiments. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and communicate openly with the public (see Figure 1). Wiki projects can be used to develop disciplinary knowledge and social and communication skills by making creative use of the Wikimedia Foundation wiki projects as freely available, active, open knowledge, collaborative online technology platforms.

Participants (staff, students, others) contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. The work is immediately available to the public and can be edited by anyone. In this way the open educational resources are iteratively improved. The radical transparency of open editing can intially be challenging for university staff and students but ultimately this approach empowers participants' capacity and confidence in contributing directly to the knowledge commons and optimises the utility of the work.

Wiki-based assignment formats can vary depending on:
* Subject area (e.g., humanities, sciences, professional fields)
* Level of study (e.g., informal, K-12, undergraduate, postgraduate, professional learning)
* Intended learning outcomes and development of generic skills (e.g., critical thinking, communication, digital literacy)

Typical tasks involve creating, curating, or improving open educational resources such as:
* Books (e.g., open textbooks) — Wikibooks
* Data (e.g., linked data items and data analysis) — Wikidata
* Conferences (e.g., home page, applications, abstracts, program, presentations) — Wikiversity
* Encyclopedic articles — Wikipedia
* Essays and analytical reports — Wikiversity
* Images, audio, and video (with metadata and licensing) — Wikimedia Commons
* Fact sheets and study guides — Wikiversity
* Journal-style articles and literature reviews — Wikiversity
* Learning activities (e.g., lectures, tutorials, workshops) — Wikiversity
* Manuals, tutorials, and how-to guides — Wikiversity

Open educational wikis can serve as [[w:Content management system|content management systems]] for hosting teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. Wikis can support the development of open textbooks. Compared to popular institutionally supported textbook platforms for open textbooks such as [[w:PressBooks|PressBooks]], there are no fees to use Wikimedia projects and they enable more diverse, collaborative, and participatory knowledge production.&lt;ref group=&quot;note&quot;&gt;In the context of [[w:Tertiary education in Australia|Australian higher education]], platforms such as PressBooks are typically staff-controlled, with limited opportunities for student authorship and co-creation.&lt;/ref&gt;

A notable feature of wikis is that every page has a complete, searchable edit history. Each revision can be reviewed and, if necessary, reverted, ensuring that no content is permanently lost. Most edits incrementally improve the quality of a page; however, a small proportion are unhelpful and are therefore undone. As a rough guide, approximately 95% of edits are retained, while around 5% are reverted or deleted.

Wikis support long-term knowledge preservation. For students, an open wiki assignment ensures that their work remains publicly available and that they retain access to course materials  materials beyond graduation. For staff, access to open wiki materials continues regardless of institutional affiliation.

On wikis, disagreements about content are addressed through open discussion and consensus-building. This creates a distinctive collaborative environment in which students develop core skills in argumentation, communication, and negotiation. Wiki content can also be readily forked, similar to open-source software, enabling alternative versions to evolve in parallel. There is also a need for translation and development of open knowledge materials in different languages. As a result, wikis foster a pragmatic, solution-focused culture for collaborative knowledge development.

==Wikimedia projects==
===Wikipedia===
The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists.

The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. 

However, I would cast the net wider than Wikipedia because:
* Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence.
* Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources.

For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects.

===Beyond Wikipedia===

Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment.

{{sisterprojects}}
Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development.

An open wiki higher education assignment generally involves:
* Students contributing discipline-relevant content to the global knowledge commons via one or more of the Wikimedia Foundation sister projects 
* Assignment tasks centre on producing and refining useful knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences
* Outputs can include text, media, data, and learning resources  
* Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages])  

Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources.

{| class=&quot;wikitable&quot; style=&quot;margin: 0 auto;&quot;
|+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments
! Project
! Purpose
! Example assignments
|-
| [[b:|Wikibooks]]
| New books (e.g., textbooks)
| 
* Contribute to development of an open textbook
* Curate and improve existing OER book chapters
* Package a series of related articles into a new book
|-
| [[commons:|Wikimedia Commons]]
| Images, audio, and video
|
* Contribute high-quality educational media
* Improve metadata and categorisation
* Create educational diagrams and illustrations
* Upload field recordings or interviews
|-
| [[d:Wikidata|Wikidata]]
| Structured, linked open data
|
* Create and curate datasets
* Link concepts across Wikimedia projects
* Model relationships between entities
* Support data-driven research and analysis
|-
| [[q:|Wikiquote]]
| Quotations
|
* Curate and improve text quotes from primary sources such as political speeches
* Create categories for quotes by theme or topic
* Add citations and verification to existing quotes
|-
| [[species:|Wikispecies]]
| Taxonomy and species classification
|
* Curate and improve taxonomic entries for species
* Add citations for classification and nomenclature
* Contribute information about newly described species
* Improve links between species and related Wikimedia projects
|-
| [[s:Main Page|Wikisource]]
| Primary texts and historical documents
|
* Transcribe and proofread source texts
* Annotate and contextualise historical documents
* Curate thematic collections of primary sources
|-
| [[v:|Wikiversity]]
| Learning, teaching, and research
|
* Create open educational resources
* Develop teaching materials (e.g., lesson plans, self-assessment quizzes)
* Publish student research project summaries
* Improve existing learning resources by adding new text and multimedia
|-
| [[voy:Main Page|Wikivoyage]]
| Travel guides and geographic knowledge
|
* Develop place-based guides (e.g., regions, cities)
* Contribute cultural, historical, or environmental information
* Integrate fieldwork or experiential learning outputs
|-
| [[wikt:Main Page|Wiktionary]]
| Lexical and linguistic resources
|
* Create and refine dictionary entries
* Analyse word meanings, usage, and etymology
* Contribute multilingual translations and examples
|-
| [[w:|Wikipedia]]
| Encyclopedic information
|
* Contribute to articles related to the class topic where a gap exists
* Improve the quality and accuracy of existing articles
* Add citations and references to unverified text
* Curate and improve a category of articles related to a specific subject area
|}

==Open wiki assignments==
Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available.

Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include:
* '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s).
* '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators.
* '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions.
* '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]].
*'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. 
* '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research.

Advantages of open wiki assignments include:
* '''Perpetuity''' - ongoing availability of resources
* '''Linkability''' - cross-linking of projects and external resources
* '''Editability''' - resources can be improved by anyone
* '''Discussability''' - each resource has a discussion page
* '''Showability''' - resources showcase curator skills and knowledge
* '''Transparency''' - resource edit history and can be reviewed
* '''Forkability''' - open licence allows development of alternative resources

==Examples==
Here are some examples of open wiki assignments:
* [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia 
* [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia
* [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA

Whilst not assignments per se, these innovative open wiki resources may inspire:
* [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries

==Activities==
* Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]]
* Edit your [[Help:User page|Wikiversity user page]]
* Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]]
* Brainstorm what you or your students could contribute
* Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support

==Bio==
[[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''.

==See also==
;Wikimedia Foundation
* [[meta:Education|Education]] (Global WMF hub)
* [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia)

;Wikipedia
* [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta)

;Wikiversity
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article)
* [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page)
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation)

[[Category:User:Jtneill/Presentations/Open education]]
[[Category:User:Jtneill/Presentations/Wikiversity]]

== Notes ==
&lt;references group=&quot;note&quot; /&gt;&lt;!--
== References ==
&lt;references /&gt;
--&gt;</text>
      <sha1>9627slg4v57j56yrhxuc4jwl7uz41ra</sha1>
    </revision>
    <revision>
      <id>2804466</id>
      <parentid>2804465</parentid>
      <timestamp>2026-04-12T11:09:57Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Wikimedia projects */ + Intro paragraph</comment>
      <origin>2804466</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19174" sha1="qxr9wce1g0l7k4bbcnl73ufxjokwh0h" xml:space="preserve">{{title|Open wiki assignments for authentic learning}}
&lt;div style=&quot;text-align: center&quot;&gt;

[[User:Jtneill|James T. Neill]]&lt;br&gt;
[[v:University of Canberra|University of Canberra]]

[https://educationexpress.uts.edu.au/blog/2026/03/31/join-us-at-open-education-week-2026/ Open Education Week 2026, University of Technology Sydney]&lt;br&gt;
Friday 24 April, 2026 11:00 - 12:00 AEST

[https://utsmeet.zoom.us/j/84179400467 Zoom link]

&lt;!-- Slides TBA (Google)&lt;br&gt;
Video TBA (YouTube; 53:37 mins including Q&amp;A)
[https://www.linkedin.com/feed/update/urn:li:activity:7174278714963230720/ (example) LinkedIn post]
[https://twitter.com/jtneill/status/1768516693553565884 (example) X post]
--&gt;
&lt;/div&gt;
==Overview==
{{Nutshell|Turning student assignments into meaningful, public knowledge through practical, open wiki-based assessment strategies.}}
{{RoundBoxTop|theme=3}}
Many student assignments are written for one person, read once, and then never read again. 

In this session, [[User:Jtneill|Dr. James Neill]], from the Discipline of Psychology at the [[University of Canberra]], will challenge that model by exploring how open [[w:Wiki|wiki]] assignments can turn student work into useful, open knowledge. 

Rather than producing disposable assessments, students can curate their work via [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] including [[w:Wikipedia|Wikipedia]], [[Main page|Wikiversity]], and [[c:Wiki Commons|Wiki Commons]]. Student editing of these widely used knowledge platforms helps develop their critical thinking, collaboration and communication skills, and technological literacy by writing for a real audience. Students emerge with a learning artifact they can share on social media and in their resume and eportfolio.

The session will explore: 
* What open wiki assignments look like in practice, and where they go wrong
* The realities of working in publicly editable spaces (including having work changed or deleted)
* Practical strategies and supports for getting started, including account creation, editing a user page, and finding your way around

This session is for tertiary educators who are curious about [[w:Open education|open education]] using wikis but may be sceptical, short on time, or wary of adding complexity to their teaching.
{{RoundBoxBottom}}

==Introduction==
[[w:Wiki|Wiki]]s were developed in the early days of the world wide web (1994) as the simplest web page that anyone can edit (for more, see [[w:History of wikis|history of wikis]]). Today, wikis are the foundation for the global open knowledge projects developed by the Wikimedia Foundation, the best known of which is Wikipedia. These dozen or so sister wiki projects have in common the lofty goal of making the sum of all human knowledge freely available to all—and allowing that knowledge to be editable by anyone. These projects have a high impact because they are quickly listed by search engines, often visited, updatable, used to train artificial intelligence, and openly licensed to allow use by others.

[[File:Wiki project case study onion diagram.svg|right|thumb|'''Figure 1'''. An onion layer model of open wiki assignments for authentic learning]]
Wiki technology enables grand social experiments. Like universities, wikis are great places for staff and students to hang out, collaborate and engage in learning and research activities, and communicate openly with the public (see Figure 1). Wiki projects can be used to develop disciplinary knowledge and social and communication skills by making creative use of the Wikimedia Foundation wiki projects as freely available, active, open knowledge, collaborative online technology platforms.

Participants (staff, students, others) contribute wiki content under open licenses ([https://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Share Alike]) and collaborate by editing and commenting on each other’s work. The work is immediately available to the public and can be edited by anyone. In this way the open educational resources are iteratively improved. The radical transparency of open editing can intially be challenging for university staff and students but ultimately this approach empowers participants' capacity and confidence in contributing directly to the knowledge commons and optimises the utility of the work.

Wiki-based assignment formats can vary depending on:
* Subject area (e.g., humanities, sciences, professional fields)
* Level of study (e.g., informal, K-12, undergraduate, postgraduate, professional learning)
* Intended learning outcomes and development of generic skills (e.g., critical thinking, communication, digital literacy)

Typical tasks involve creating, curating, or improving open educational resources such as:
* Books (e.g., open textbooks) — Wikibooks
* Data (e.g., linked data items and data analysis) — Wikidata
* Conferences (e.g., home page, applications, abstracts, program, presentations) — Wikiversity
* Encyclopedic articles — Wikipedia
* Essays and analytical reports — Wikiversity
* Images, audio, and video (with metadata and licensing) — Wikimedia Commons
* Fact sheets and study guides — Wikiversity
* Journal-style articles and literature reviews — Wikiversity
* Learning activities (e.g., lectures, tutorials, workshops) — Wikiversity
* Manuals, tutorials, and how-to guides — Wikiversity

Open educational wikis can serve as [[w:Content management system|content management systems]] for hosting teaching and learning materials beyond the closed environments of institutional [[w:Learning management system|learning management systems]]. Wikis can support the development of open textbooks. Compared to popular institutionally supported textbook platforms for open textbooks such as [[w:PressBooks|PressBooks]], there are no fees to use Wikimedia projects and they enable more diverse, collaborative, and participatory knowledge production.&lt;ref group=&quot;note&quot;&gt;In the context of [[w:Tertiary education in Australia|Australian higher education]], platforms such as PressBooks are typically staff-controlled, with limited opportunities for student authorship and co-creation.&lt;/ref&gt;

A notable feature of wikis is that every page has a complete, searchable edit history. Each revision can be reviewed and, if necessary, reverted, ensuring that no content is permanently lost. Most edits incrementally improve the quality of a page; however, a small proportion are unhelpful and are therefore undone. As a rough guide, approximately 95% of edits are retained, while around 5% are reverted or deleted.

Wikis support long-term knowledge preservation. For students, an open wiki assignment ensures that their work remains publicly available and that they retain access to course materials  materials beyond graduation. For staff, access to open wiki materials continues regardless of institutional affiliation.

On wikis, disagreements about content are addressed through open discussion and consensus-building. This creates a distinctive collaborative environment in which students develop core skills in argumentation, communication, and negotiation. Wiki content can also be readily forked, similar to open-source software, enabling alternative versions to evolve in parallel. There is also a need for translation and development of open knowledge materials in different languages. As a result, wikis foster a pragmatic, solution-focused culture for collaborative knowledge development.

==Wikimedia projects==
This section describes the nature of Wikipedia student assignments, including what to be wary about. It then explains how open wiki assignments can be conducted on other Wikimedia Foundation sister projects which arguably offer more targtted, specific environments depending on the discipline and learning objectives. Wikiversity is the main project emphasises because its [[Wikiversity:Mission|mission]] is most closely aligned with [[w:higher education|higher education]].

===Wikipedia===
The most successful and notable educational wiki projects are supported by the [[w:Wikimedia Foundation|Wikimedia Foundation]]. [[w:Wikipedia|Wikipedia]] is the best known. Many university subjects use assignments which involve students contributing to Wikipedia articles related to the class topic and where an encyclopedic gap or need exists.

The best known Wikipedia assignments are facilitated by the [[w:Wiki Education Foundation|Wiki Education Foundation]], a separate non-profit entity which supports Canadian and U.S. college faculty and postsecondary institutions to undertake such Wikipedia assignments with their students. Non-U.S./Canadian instituations can conduct similar assignments on their own. 

However, I would cast the net wider than Wikipedia because:
* Wikipedia editing, especially for newcomers, isn't for the faint-hearted. Imagine taking a group of learner drivers into a busy central business district at peak hour for their first lesson. As the most popular and populated wiki, Wikipedia can be a crowded editing space, making it difficult for new editors to get a foothold and gain in confidence.
* Wikipedia focuses on encyclopedic content and not on formats such as argument/debate, opinion, essays, creative work, original research, or targetted open educational resources.

For these two reasons, I encourage higher educators to also consider how their discipline, subject area, and desired learning outcomes may be achieved through student assignments on Wikimedia sister projects.

===Beyond Wikipedia===
Opportunities for students to contribute open knowledge extend beyond Wikipedia to the broader [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia Foundation sister projects]]. These platforms provide authentic, public-facing environments for producing, curating, and sharing openly licensed scholarly work as part of higher education assessment.

{{sisterprojects}}
Table 1 outlines how a range of sister projects can be used for student assignments, including [[w:|Wikipedia]], [[b:|Wikibooks]], [[commons:|Wikimedia Commons]], [[q:|Wikiquote]], [[species:|Wikispecies]], and [[v:|Wikiversity]]. Collectively, these support diverse forms of knowledge production, from encyclopaedic writing and open textbooks to media creation, quotation curation, taxonomic documentation, and learning resource development.

An open wiki higher education assignment generally involves:
* Students contributing discipline-relevant content to the global knowledge commons via one or more of the Wikimedia Foundation sister projects 
* Assignment tasks centre on producing and refining useful knowledge or resources—through creating, improving, curating, synthesising, verifying, linking, and communicating content for real-world audiences
* Outputs can include text, media, data, and learning resources  
* Work is openly accessible, reusable, and can be multilingual (see [https://wikiversity.org Wikiversity languages])  

Together, these platforms support a wide range of assessment formats aligned with open educational practice, including open textbooks, datasets, media artefacts, encyclopedic entries, and research-informed learning resources.

{| class=&quot;wikitable&quot; style=&quot;margin: 0 auto;&quot;
|+ Table 1. How Wikimedia Sister Projects Could Be Used for Higher Education Student Assignments
! Project
! Purpose
! Example assignments
|-
| [[b:|Wikibooks]]
| New books (e.g., textbooks)
| 
* Contribute to development of an open textbook
* Curate and improve existing OER book chapters
* Package a series of related articles into a new book
|-
| [[commons:|Wikimedia Commons]]
| Images, audio, and video
|
* Contribute high-quality educational media
* Improve metadata and categorisation
* Create educational diagrams and illustrations
* Upload field recordings or interviews
|-
| [[d:Wikidata|Wikidata]]
| Structured, linked open data
|
* Create and curate datasets
* Link concepts across Wikimedia projects
* Model relationships between entities
* Support data-driven research and analysis
|-
| [[q:|Wikiquote]]
| Quotations
|
* Curate and improve text quotes from primary sources such as political speeches
* Create categories for quotes by theme or topic
* Add citations and verification to existing quotes
|-
| [[species:|Wikispecies]]
| Taxonomy and species classification
|
* Curate and improve taxonomic entries for species
* Add citations for classification and nomenclature
* Contribute information about newly described species
* Improve links between species and related Wikimedia projects
|-
| [[s:Main Page|Wikisource]]
| Primary texts and historical documents
|
* Transcribe and proofread source texts
* Annotate and contextualise historical documents
* Curate thematic collections of primary sources
|-
| [[v:|Wikiversity]]
| Learning, teaching, and research
|
* Create open educational resources
* Develop teaching materials (e.g., lesson plans, self-assessment quizzes)
* Publish student research project summaries
* Improve existing learning resources by adding new text and multimedia
|-
| [[voy:Main Page|Wikivoyage]]
| Travel guides and geographic knowledge
|
* Develop place-based guides (e.g., regions, cities)
* Contribute cultural, historical, or environmental information
* Integrate fieldwork or experiential learning outputs
|-
| [[wikt:Main Page|Wiktionary]]
| Lexical and linguistic resources
|
* Create and refine dictionary entries
* Analyse word meanings, usage, and etymology
* Contribute multilingual translations and examples
|-
| [[w:|Wikipedia]]
| Encyclopedic information
|
* Contribute to articles related to the class topic where a gap exists
* Improve the quality and accuracy of existing articles
* Add citations and references to unverified text
* Curate and improve a category of articles related to a specific subject area
|}

==Open wiki assignments==
Developing reusable assignments on the web rather than disposable assignments (which are written and read once) means that the value of student work is recognised and realised beyond the purpose of gaining academic credit. Instead of being tossed into the learning management system assignment dumpster and never seen again, students' learning artifacts can be live and publicly available.

Given that normative nature of disposable assignments in higher education, the idea of renewable, online, public assessment can seem oddly confronting. Some common reactions (from educators and students) include:
* '''What if someone changes my work?''' - Hopefully they improve it; otherwise, simply revert the edit(s).
* '''What if someone vandalises my work?''' - This is rare and is typically detected and corrected quickly by bots and administrators.
* '''What if someone deletes my work?''' - All edits are preserved in the version history, making it straightforward to restore earlier versions.
* '''Editing the internet is scary and I do not know how to do it.''' - Basic wiki editing skills can be learned in a [[Motivation and emotion/Tutorials/Wiki editing|1-hour tutorial]].
*'''What if I don't want my work on the internet?''' - Students own the copyright to their work and must opt in to sharing it. They also have a right to privacy. Provide an alternative task or submission format(s) so that students can achieve the assignment's learning outcomes without putting their work on the open internet. 
* '''Open wikis seems like a copyright nightmare. My institution would never allow staff to contribute teaching materials openly.''' - Institutional policies may require negotiation or adaptation to support open educational resource sharing. However, students typically retain copyright over their work and may choose to share it under an open licence. Where this is not appropriate, alternative assessment options can be provided. Open educational practices are increasingly adopted in Australian universities, similar to the earlier expansion of [[w:Open access|open access]] in research.

Advantages of open wiki assignments include:
* '''Perpetuity''' - ongoing availability of resources
* '''Linkability''' - cross-linking of projects and external resources
* '''Editability''' - resources can be improved by anyone
* '''Discussability''' - each resource has a discussion page
* '''Showability''' - resources showcase curator skills and knowledge
* '''Transparency''' - resource edit history and can be reviewed
* '''Forkability''' - open licence allows development of alternative resources

==Examples==
Here are some examples of open wiki assignments:
* [[b:Exercise as it relates to Disease|Exercise as it relates to disease]] - exercise physiology students write 1,000-word article critiques (Wikibooks), Faculty of Health, University of Canberra, Australia 
* [[Motivation and emotion/Book|Motivation and emotion]] - undergraduate psychology students write 3,000-word online book chapters about unique topics (Wikiversity), Faculty of Health, University of Canberra, Australia
* [[Digital Media Concepts|Digital artists wiki project assignment]] (Wikiversity) - Multimedia Department, Ohlone College, CA, USA

Whilst not assignments per se, these innovative open wiki resources may inspire:
* [[Global Audiology]] - collaboratively developed open wiki portal enabling international, student-contributable knowledge on audiology practice to address inequities in hearing care access, particularly in low- and middle-income countries

==Activities==
* Create a [[Wikiversity:Why create an account|global Wikimedia Foundation user account]]
* Edit your [[Help:User page|Wikiversity user page]]
* Explore available Wikiversity resources: [[Special:Search|Search]], [[Portal:Learning Projects|Portals]], [[Help:Guides|Tours]]
* Brainstorm what you or your students could contribute
* Visit the [[Wikiversity:Colloquium|Colloquium]] and [[Wikiversity:Staff|Wikiversity staff]] so you know where to get support

==Bio==
[[User:Jtneill|James Neill]] is an Assistant Professor in the Discipline of Psychology, Faculty of Health, [[University of Canberra]]. He is a proponent of open educational practices and contributes [[Open Educational Resources|open educational resources]] via open wiki platforms. James is an [[Main page|English Wikiversity]] [[Wikiversity:Custodianship|custodian]] and [[Wikiversity:Bureaucratship|bureaucrat]] who has made over 80,000 edits since 2005. Learn more about James' ''[[User:Jtneill/Teaching/Philosophy|teaching philosophy]]''.

==See also==
;Wikimedia Foundation
* [[meta:Education|Education]] (Global WMF hub)
* [[w:Wikipedia:Wikimedia_sister_projects|Wikimedia sister projects]] (Wikipedia)

;Wikipedia
* [[meta:Wiki Education Foundation|Wiki Education Foundation]] (meta)

;Wikiversity
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (article)
* [[Wikiversity:Who are Wikiversity participants?|Who are Wikiversity participants?]] (page)
* [[User:Jtneill/Presentations/Wikis in open education: A psychology case study|Wikis in open education: A psychology case study]] (presentation)

[[Category:User:Jtneill/Presentations/Open education]]
[[Category:User:Jtneill/Presentations/Wikiversity]]

== Notes ==
&lt;references group=&quot;note&quot; /&gt;&lt;!--
== References ==
&lt;references /&gt;
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    <revision>
      <id>2804390</id>
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      <sha1>fuunhib8gf2i4rd6pgmdrixxeavjn7k</sha1>
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  <page>
    <title>Portal:Bikol</title>
    <ns>102</ns>
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    <revision>
      <id>2804290</id>
      <parentid>2803934</parentid>
      <timestamp>2026-04-11T12:05:04Z</timestamp>
      <contributor>
        <username>ShakespeareFan00</username>
        <id>6645</id>
      </contributor>
      <minor />
      <origin>2804290</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4185" sha1="fcjmypod7fbwp5jmlus18fq75boy3bs" xml:space="preserve">&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:99%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;margin-top:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Philippine Languages Department&lt;/h2 &gt;
[[Image:Globe of letters.svg|right|88px|Languages]]
Welcome to the '''Bikol Department''' at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]].

Bikol is an Austronesian language spoken in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca.
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Courses&lt;/h2 &gt;
[[Image:Crystal128-kanagram.svg|right|44px|]]
* [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol.&lt;br&gt;
''Province focus: Albay''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol 2|Bikol 2]] - get to know the alphabets and how they are pronounced in Bikol.&lt;br&gt;
''Province focus: Sorsogon''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol 3|Bikol 3]] - learn how time is being said in Bikol.&lt;br&gt;''Province focus: Catanduanes''&lt;/br&gt;

&lt;div style=&quot;display:block;width:99%;float:left&quot;&gt;
&lt;div style=&quot;width:50%;display:block;float:left;&quot;&gt;
&lt;div&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol.&lt;br&gt;''Province focus: Camarines Sur''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Grammar|Grammar]] - this lesson will answer your curiosity on how words are being joined together in Bikol.&lt;br&gt;''Province focus: Camarines Norte''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Quiz|Quiz]] - challenge yourself on how well you familiarize the grammar in Bikol.&lt;br&gt;''Province focus: Masbate''&lt;/br&gt;&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Division News&lt;/h2 &gt;
[[Image:Nuvola apps knewsticker.png|right|50px|]]* '''April 8, 2026 ''' - Department founded!&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;
&lt;div&gt; 
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;See also&lt;/h2 &gt;
[[Image:Nuvola apps bookcase.svg|right|44px|]]
[[Southeast Asian Languages/Philippine Languages/Bikol]]&lt;/div&gt;

&lt;div style=&quot;float: right; margin: 2px;&quot;&gt;
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
|{{center|'''Additional Wikimedia resources'''}}
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:b:Bikol|Textbook]] at Wikibooks
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | {{w|Central Bikol|Article}} at Wikipedia
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:b:Wikivoyage|Bikol phrasebook]]  at Wikivoyage
|}&lt;/div&gt;


&lt;/div&gt;

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&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Philippine Languages Department&lt;/h2 &gt;
[[Image:Globe of letters.svg|right|88px|Languages]]
Welcome to the '''Bikol Department''' at Wikiversity, part of the [[Portal:Foreign Language Learning|Center for Foreign Language Learning]] and the [[School:Language and Literature|School of Language and Literature]].

Bikol is an Austronesian language spoken in the Philippines particularly on the Bicol Peninsula in the island of Luzon. Standard Bikol is based on the dialect of Naga City and is spoken in a wide area stretching from Camarines Norte, most of Camarines Sur, the entire east coast of Albay (including Legazpi City and Tabaco City) and northern Sorsogon. Standard Bikol is generally understood by other Bikol speakers and is the regional lingua franca.
&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Courses&lt;/h2 &gt;
[[Image:Crystal128-kanagram.svg|right|44px|]]
* [[Southeast Asian Languages/Philippine Languages/Bikol 1|Bikol 1]] - you will learn how to greet and introduce yourself in Bikol.&lt;br&gt;
''Province focus: Albay''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol 2|Bikol 2]] - get to know the alphabets and how they are pronounced in Bikol.&lt;br&gt;
''Province focus: Sorsogon''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol 3|Bikol 3]] - learn how time is being said in Bikol.&lt;br&gt;''Province focus: Catanduanes''&lt;/br&gt;

&lt;div style=&quot;display:block;width:99%;float:left&quot;&gt;
&lt;div style=&quot;width:50%;display:block;float:left;&quot;&gt;
&lt;div&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Vocabulary|Vocabulary]] - this lesson will give you some useful words and phrases in Bikol.&lt;br&gt;''Province focus: Camarines Sur''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Grammar|Grammar]] - this lesson will answer your curiosity on how words are being joined together in Bikol.&lt;br&gt;''Province focus: Camarines Norte''&lt;/br&gt;
* [[Southeast Asian Languages/Philippine Languages/Bikol/Lesson:Quiz|Quiz]] - challenge yourself on how well you familiarize the grammar in Bikol.&lt;br&gt;''Province focus: Masbate''&lt;/br&gt;&lt;/div&gt;

&lt;div style=&quot;display:block;border:1px solid #aaaaaa;vertical-align: top;width:100%; background-color:#f9f9ff; {{Text color default}};margin-bottom:10px;padding-bottom:5px;padding-left:5px;padding-right:4px;&quot;&gt;
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;Division News&lt;/h2 &gt;
[[Image:Nuvola apps knewsticker.png|right|50px|]]* '''April 8, 2026 ''' - Department founded!&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;div&gt; 
&lt;h2 style=&quot;padding:3px; background:#aaccff; {{Text color default}}; color:#000; text-align:center; font-weight:bold; font-size:100%; margin-bottom:5px;margin-top:0;margin-left:-5px;margin-right:-4px;&quot;&gt;See also&lt;/h2 &gt;
[[Image:Nuvola apps bookcase.svg|right|44px|]]
[[Southeast Asian Languages/Philippine Languages/Bikol]]&lt;/div&gt;

&lt;div style=&quot;float: right; margin: 2px;&quot;&gt;
{| class=&quot;wikitable&quot; border=&quot;1&quot;
|-
|{{center|'''Additional Wikimedia resources'''}}
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:b:Bikol|Textbook]] at Wikibooks
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | {{w|Central Bikol|Article}} at Wikipedia
|-
! style=&quot;background: #lime&quot; colspan=&quot;6&quot; | [[:b:Wikivoyage|Bikol phrasebook]]  at Wikivoyage
|}&lt;/div&gt;


&lt;/div&gt;

__NOEDITSECTION__
__NOTOC__

[[Category:Philippine languages]]</text>
      <sha1>6wd7shgqsspnp15w6ek4zpkt5jykqrw</sha1>
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    <title>Topic:Southeast Asian Languages/Contributors and Students</title>
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      <text bytes="126" sha1="tl796j4oktcd2peg04wpinw6y7jcsxk" xml:space="preserve">* [[User:CarlessParking|Karl Ian Basallote]] - I contributed my native language Bikol to [[Portal:Southeast Asian Languages]].</text>
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      <timestamp>2026-04-11T12:00:31Z</timestamp>
      <contributor>
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      <text bytes="126" sha1="r875h3ojmc4rj1vgz2n7svwavatye40" xml:space="preserve">* [[User:CarlessParking|Karl Ian Basallote]] - I contributed my native language Bikol to [[Portal:Southeast Asian languages]].</text>
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      <text bytes="103" sha1="p558nslxbq3bqqnbd71ajt5sk8errre" xml:space="preserve">I'm a Bicolano from Bicol Region specifically from the province of Albay.

Karl Ian Basallote y Basilla</text>
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      <comment>Created page with &quot;==Family law== In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed...&quot;</comment>
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      <text bytes="867" sha1="1ao60q3gam9ehojidpmhmy9c483yumt" xml:space="preserve">==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;</text>
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      <text bytes="883" sha1="c8iryo7oy24tde3vrak9iyqm4p0ifys" xml:space="preserve">==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

==References==</text>
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{{Comparative law and justice project|region=Asia}}

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|Flag of the Philippines]]

==Basic information==
The Philippines archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|right|thumb|250px|Flag of the Philippines]]

==Basic information==
The Philippines archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president]] is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the [[separation of Church and State]] exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==</text>
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{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president]] is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==</text>
      <sha1>m25t06j8tag8qf74q2a06jqwyr4tzof</sha1>
    </revision>
    <revision>
      <id>2804311</id>
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      <timestamp>2026-04-11T12:38:23Z</timestamp>
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      <text bytes="13757" sha1="mmlcm3azv76ovbl0dn01tn7xw62429m" xml:space="preserve">Part of the [[Comparative law and justice]] Wikiversity Project
{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==</text>
      <sha1>mmlcm3azv76ovbl0dn01tn7xw62429m</sha1>
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    <revision>
      <id>2804312</id>
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      <minor />
      <comment>added [[Category:Philippines]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="13783" sha1="chm9dw5fwmt4uorvn3i8axssvlbts6r" xml:space="preserve">Part of the [[Comparative law and justice]] Wikiversity Project
{{Comparative law and justice project|region=Asia}}

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)
{{law}}

==Basic information==
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)
{{law}}

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC){{law}}

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==References==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $[[Mexican peso#First peso|MXN]]800,000{{inflation|US|1|1897|r=2}}/2 round 2}} today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par.}} and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the [[Philippine five peso bill]]]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000{{inflation|US|1|1897|r=2}}/2 round 2}} today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2}} today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Family law==
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

==Adoption==
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the [[coastal zone]].&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in [[Metro Manila]]. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; [[Income inequality in the Philippines|The Philippines' income inequality]] began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were [[Tagalog people|Tagalog]] (26.0 percent), [[Visayans]] [excluding the [[Cebuano people|Cebuano]], [[Hiligaynon people|Hiligaynon]], and [[Waray people|Waray]]] (14.3 percent), [[Ilocano people|Ilocano]] and Cebuano (both eight percent), Hiligaynon (7.9 percent), [[Bicolano people|Bikol]] (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The [[Indigenous peoples of the Philippines|country's indigenous peoples]] consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the [[Igorot people|Igorot]], [[Lumad]], [[Mangyan]], and the [[Peoples of Palawan|indigenous peoples of Palawan]].&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

[[Negrito]]s are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an [[Australo-Melanesian|Australoid]] group, a remnant of the [[Southern Dispersal|first human migration from Africa to Australia]] who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a [[Denisovan]] admixture in their [[genome]].&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as [[Austronesian peoples|Austronesians]] speaking [[Malayo-Polynesian languages]].&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

The Austronesian population's origin is uncertain, but relatives of [[Taiwanese indigenous peoples|Taiwanese aborigines]] probably brought their language and mixed with the region's existing population.&lt;ref&gt;{{cite journal |last1=Capelli |first1=Christian |last2=Wilson |first2=James F. |last3=Richards |first3=Martin |last4=Stumpf |first4=Michael P.H. |last5=Gratrix |first5=Fiona |last6=Oppenheimer |first6=Stephen |last7=Underhill |first7=Peter |last8=Ko |first8=Tsang-Ming |year=2001 |title=A Predominantly Indigenous Paternal Heritage for the Austronesian-Speaking Peoples of Insular South Asia and Oceania |url=http://hpgl.stanford.edu/publications/AJHG_2001_v68_p432.pdf |journal=[[American Journal of Human Genetics]] |volume=68 |issue=2 |pages=432–443 |doi=10.1086/318205 |pmc=1235276 |pmid=11170891 |archive-url=https://web.archive.org/web/20110511201051/http://hpgl.stanford.edu/publications/AJHG_2001_v68_p432.pdf |archive-date=May 11, 2011 |access-date=December 18, 2009 |doi-access=free}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Soares |first1=Pedro A. |last2=Trejaut |first2=Jean A. |last3=Rito |first3=Teresa |last4=Cavadas |first4=Bruno |last5=Hill |first5=Catherine |last6=Eng |first6=Ken Khong |last7=Mormina |first7=Maru |last8=Brandão |first8=Andreia |last9=Fraser |first9=Ross M. |last10=Wang |first10=Tse-Yi |last11=Loo |first11=Jun-Hun |last12=Snell |first12=Christopher |last13=Ko |first13=Tsang-Ming |last14=Amorim |first14=António |last15=Pala |first15=Maria |last16=Macaulay |first16=Vincent |last17=Bulbeck |first17=David |last18=Wilson |first18=James F. |last19=Gusmão |first19=Leonor |last20=Pereira |first20=Luísa |last21=Oppenheimer |first21=Stephen |last22=Lin |first22=Marie |last23=Richards |first23=Martin B. |title=Resolving the ancestry of Austronesian-speaking populations |journal=[[Human Genetics (journal)|Human Genetics]] |publisher=[[Springer Science+Business Media]] |date=March 2016 |volume=135 |issue=3 |pages=309–326 |doi=10.1007/s00439-015-1620-z |pmc=4757630 |pmid=26781090 |doi-access=free}}&lt;/ref&gt; The Lumad and [[Sama-Bajau]] ethnic groups have an ancestral affinity with the [[Austroasiatic languages|Austroasiatic]] and [[Mlabri language|Mlabri-speaking]] [[Lua people|Htin]] peoples of mainland Southeast Asia. Westward expansion of [[Papuan languages|Papuan ancestry]] from [[Papua New Guinea]] to eastern Indonesia and [[Mindanao]] has been detected in the [[Blaan people]] and the [[Sangir language]], while ancient immigration added some [[India]]n ancestry to the precolonial [[History of the Philippines (900–1565)|Indianized kingdoms]] of the islands.&lt;ref&gt;{{cite journal |last1=Larena |first1=Maximilian |last2=Sanchez-Quinto |first2=Federico |last3=Sjödin |first3=Per |last4=McKenna |first4=James |last5=Ebeo |first5=Carlo |last6=Reyes |first6=Rebecca |last7=Casel |first7=Ophelia |last8=Huang |first8=Jin-Yuan |last9=Hagada |first9=Kim Pullupul |last10=Guilay |first10=Dennis |last11=Reyes |first11=Jennelyn |date=March 30, 2021 |title=Multiple migrations to the Philippines during the last 50,000 years |journal=[[Proceedings of the National Academy of Sciences]] |publisher=[[National Academy of Sciences]] |volume=118 |issue=13 |article-number=e2026132118 |bibcode=2021PNAS..11826132L |doi=10.1073/pnas.2026132118 |issn=0027-8424 |pmc=8020671 |pmid=33753512 |doi-access=free}}&lt;/ref&gt;

Immigrants arrived in the Philippines from elsewhere in the Spanish Empire, especially [[Latin American Asian|from the Spanish Americas]].&lt;ref&gt;{{cite journal |last=Mawson |first=Stephanie J. |date=June 15, 2016 |title=Convicts or Conquistadores? Spanish Soldiers in the Seventeenth-Century Pacific |url=https://academic.oup.com/past/article/232/1/87/1752419 |journal=[[Past &amp; Present (journal)|Past &amp; Present]] |publisher=[[Oxford University Press]] |issue=232 |pages=87–125 |doi=10.1093/pastj/gtw008 |archive-url=https://web.archive.org/web/20180603111934/https://academic.oup.com/past/article/232/1/87/1752419 |archive-date=June 3, 2018 |access-date=July 28, 2020 |doi-access=free}}&lt;/ref&gt;&lt;ref name=&quot;Mehl-2016&quot;&gt;{{cite book |last=Mehl |first=Eva Maria |url=https://www.cambridge.org/core/books/forced-migration-in-the-spanish-pacific-world/22713BE2A688A4F8DFF62EDE85BE427E |title=Forced Migration in the Spanish Pacific World: From Mexico to the Philippines, 1765–1811 |date=2016 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-1-316-48012-0 |doi=10.1017/CBO9781316480120}}&lt;/ref&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20180614082235/https://www.cambridge.org/core/books/forced-migration-in-the-spanish-pacific-world/unruly-mexicans-in-manila/EF2599210A0715A5A91B23BB9D84B96C|name=Chpt. 6}}}}&lt;ref name= &quot;Intercolonial&quot;&gt;{{cite book |last=Park |first=Paula C. |url=https://books.google.com/books?id=Jg5cEAAAQBAJ |title=Intercolonial Intimacies: Relinking Latin/o America to the Philippines, 1898–1964 |date=2022 |publisher=[[University of Pittsburgh Press]] |location=Pittsburgh, Pa. |isbn=978-0-8229-8873-1 |language=en |chapter=3: On the Globality of Mexico and the Manila Galleon}}&lt;/ref&gt; A 2016 [[Geno 2.0 Next Generation|National Geographic]] project [[Genetic studies on Filipinos|concluded]] that people living in the Philippine archipelago carried [[genetic marker]]s in the following percentages: 53 percent [[Southeast Asia]] and [[Oceania]], 36 percent [[East Asia]], 5 percent [[Southern Europe]], 3 percent [[South Asia|Southern Asia]], and 2 percent Native American (from [[Latin America]]).&lt;ref name=&quot;Mehl-2016&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20180614082235/https://www.cambridge.org/core/books/forced-migration-in-the-spanish-pacific-world/unruly-mexicans-in-manila/EF2599210A0715A5A91B23BB9D84B96C|name=Chpt. 6}}}}&lt;ref&gt;{{#invoke:cite|web|title=Reference Populations – Geno 2.0 Next Generation |url=https://genographic.nationalgeographic.com/reference-populations-next-gen/ |archive-url=https://web.archive.org/web/20160704204736/https://genographic.nationalgeographic.com/reference-populations-next-gen/ |archive-date=July 4, 2016 |website=[[National Geographic]]}}&lt;/ref&gt;

Descendants of mixed-race couples are known as [[Filipino Mestizos|Mestizos]] or {{lang|fil|tisoy}},&lt;ref&gt;{{cite book |editor-last=McFerson |editor-first=Hazel M. |url=https://books.google.com/books?id=7FPLWmaGQpEC |title=Mixed Blessing: The Impact of the American Colonial Experience on Politics and Society in the Philippines |series=Contributions in Comparative Colonial Studies |date=2002 |publisher=[[Greenwood Publishing Group]] |location=Westport, Conn. |isbn=978-0-313-30791-1 |page=[https://books.google.com/books?id=7FPLWmaGQpEC&amp;pg=PA23 23]}}&lt;/ref&gt; which during the [[History of the Philippines (1565–1898)|Spanish colonial times]], were mostly composed of [[Sangley|Chinese mestizos]] ({{lang|es|Mestizos de Sangley}}), [[Spanish Filipino|Spanish mestizos]] ({{lang|es|Mestizos de Español}}) and the mix thereof ({{lang|es|[[Torna atrás|tornatrás]]}}).&lt;ref&gt;{{cite thesis |degree=PhD |last1=Villaraza |first1=Lily Ann B. |title=Yesterday, Today, and Tomorrow: A Study of Aurelio Tolentino's Articulation of Nationalism and Identity through Theatre in the Philippines during the American Colonial Period |url=https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/6759/ |website=Huskie Commons |publisher=[[Northern Illinois University]] |access-date=July 24, 2023 |archive-url=https://web.archive.org/web/20230724162046/https://huskiecommons.lib.niu.edu/cgi/viewcontent.cgi?article=7758&amp;context=allgraduate-thesesdissertations |archive-date=July 24, 2023 |pages=52–54 |date=January 1, 2017 |oclc=1257957511}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|web|date=December 8, 2015 |title=Sheer Realities: A Celebratio


'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

===Brief history===
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;

[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the [[coastal zone]].&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in [[Metro Manila]]. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; [[Income inequality in the Philippines|The Philippines' income inequality]] began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
** Bikol
* Cebuano
* Chavacano
* Hiligaynon
* Ibanag
* Ilocano
* Ivatan
* Kapampangan
* Kinaray-a
* Maguindanao
* Maranao
* Pangasinan
* Sambal
* Surigaonon
* Tagalog
* Tausug
* Waray
* Yakan

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
=== Languages ===
{{main|Languages of the Philippines}}

&lt;!--List of top five languages ONLY is suitable for this article. The complete list is what the link to Main topic just above is for.--&gt;''[[Ethnologue]]'' lists 186 languages for the Philippines, 182 of which are [[living language]]s. The other four no longer have any known speakers. Most native languages are part of the [[Philippine languages|Philippine branch]] of the [[Malayo-Polynesian languages]], which is a branch of the [[Austronesian languages|Austronesian language family]].&lt;ref name=&quot;Ethnologue-PH&quot;&gt;{{#invoke:cite|web|date=2013 |title=Philippines |url=https://www.ethnologue.com/country/PH |archive-url=https://web.archive.org/web/20130309171641/http://www.ethnologue.com/country/PH |archive-date=March 9, 2013 |access-date=February 8, 2023 |website=[[Ethnologue]] |publisher=[[SIL International]] |language=en |location=Dallas, TX}}&lt;/ref&gt; Spanish-based [[Creole language|creole]] varieties, collectively known as [[Chavacano]], are also spoken.&lt;ref&gt;{{cite book |editor-last1=Asher |editor-first1=R. E. |editor-last2=Moseley |editor-first2=Christopher |title=Atlas of the World's Languages |edition=Second |date=April 19, 2018 |publisher=[[Routledge]] |location=Florence, Ky. |isbn=978-1-317-85108-0 |url=https://books.google.com/books?id=R0xWDwAAQBAJ&amp;pg=PP226 |language=en}}&lt;/ref&gt; Many [[Philippine Negrito languages#Unique vocabulary|Philippine Negrito languages]] have unique vocabularies which survived Austronesian acculturation.&lt;ref&gt;{{cite journal |last=Reid |first=Lawrence A. |date=June 1, 1994 |title=Possible Non-Austronesian Lexical Elements in Philippine Negrito Languages |url=https://scholarspace.manoa.hawaii.edu/server/api/core/bitstreams/f88d1c43-3ab9-4d31-b1ab-d717149582e8/content |journal=[[Oceanic Linguistics]] |location=Honolulu, Hawaii |publisher=[[University of Hawaiʻi Press]] |volume=33 |issue=1 |pages=37–72 |doi=10.2307/3623000 |jstor=3623000 |archive-url=https://web.archive.org/web/20220711143411/https://scholarspace.manoa.hawaii.edu/server/api/core/bitstreams/f88d1c43-3ab9-4d31-b1ab-d717149582e8/content |archive-date=July 11, 2022 |access-date=February 18, 2023 |via=[[ScholarSpace]] |author-link1=Lawrence A. Reid |hdl=10125/32986 |hdl-access=free}}&lt;/ref&gt;

'''''Languages'''''
Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

* Aklanon
* Bikol
* Cebuano
* Chavacano
* Hiligaynon
* Ibanag
* Ilocano
* Ivatan
* Kapampangan
* Kinaray-a
* Maguindanao
* Maranao
* Pangasinan
* Sambal
* Surigaonon
* Tagalog
* Tausug
* Waray
* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
'''''Languages'''''
Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

* Aklanon
* Bikol
* Cebuano
* Chavacano
* Hiligaynon
* Ibanag
* Ilocano
* Ivatan
* Kapampangan
* Kinaray-a
* Maguindanao
* Maranao
* Pangasinan
* Sambal
* Surigaonon
* Tagalog
* Tausug
* Waray
* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

* Aklanon
* Bikol
* Cebuano
* Chavacano
* Hiligaynon
* Ibanag
* Ilocano
* Ivatan
* Kapampangan
* Kinaray-a
* Maguindanao
* Maranao
* Pangasinan
* Sambal
* Surigaonon
* Tagalog
* Tausug
* Waray
* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

Attempts to change the government to a federal, unicameral, or parliamentary government have been made since the Ramos administration.&lt;ref&gt;{{cite book |editor-last1=He |editor-first1=Baogang |url=https://books.google.com/books?id=nXf9C2xbKsYC |title=Federalism in Asia |editor-last2=Galligan |editor-first2=Brian |editor-last3=Inoguchi |editor-first3=Takashi |date=January 2009 |publisher=[[Edward Elgar Publishing]] |location=Cheltenham, England |isbn=978-1-84720-702-9 |page=[https://books.google.com/books?id=nXf9C2xbKsYC&amp;pg=PA176 176] |access-date=September 28, 2020 |archive-date=February 12, 2023 |archive-url=https://web.archive.org/web/20230212201623/https://books.google.com/books?id=nXf9C2xbKsYC |url-status=live}}&lt;/ref&gt; Philippine politics tends to be dominated by well-known families, such as political dynasties or celebrities,&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=Atun |first2=Jenna Mae L. |title=Celebrity Politics: Correlates of Voting for Celebrities in Philippine Presidential Elections |journal=Social Science Diliman |date=December 2015 |volume=11 |issue=2 |pages=1–2, 16–17 |url=http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |access-date=May 10, 2023 |publisher=[[University of the Philippines]] |language=en |issn=1655-1524 |oclc=8539228072 |archive-url=https://web.archive.org/web/20170925043652/http://journals.upd.edu.ph/index.php/socialsciencediliman/article/download/4796/4328 |archive-date=September 25, 2017}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=David |first1=Clarissa C. |last2=San Pascual |first2=Ma. Rosel S. |date=December 21, 2016 |title=Predicting vote choice for celebrity and political dynasty candidates in Philippine national elections |url=https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |journal=Philippine Political Science Journal |publisher=Philippine Political Science Association |volume=37 |issue=2 |pages=82–93 |doi=10.1080/01154451.2016.1198076 |s2cid=156251503 |access-date=January 1, 2021 |archive-date=April 17, 2021 |archive-url=https://web.archive.org/web/20210417044320/https://brill.com/view/journals/ppsj/37/2/article-p82_1.xml |url-status=live|url-access=subscription }}&lt;/ref&gt; and party switching is widely practiced.&lt;ref&gt;{{cite book |editor-last1=Hicken |editor-first1=Allen |editor-last2=Kuhonta |editor-first2=Erik Martinez |title=Party System Institutionalization in Asia: Democracies, Autocracies, and the Shadows of the Past |date=2015 |publisher=[[Cambridge University Press]] |location=New York |isbn=978-1-107-04157-8 |page=[https://books.google.com/books?id=GZmiBQAAQBAJ&amp;pg=PA316 316] |url=https://books.google.com/books?id=GZmiBQAAQBAJ |access-date=August 15, 2024 |language=en}}&lt;/ref&gt; Corruption is significant,&lt;ref&gt;{{cite journal |last1=Robles |first1=Alan C. |date=July–August 2008 |title=Civil service reform: Whose service? |url=http://www.inwent.org/ez/articles/077943/index.en.shtml |journal=[[D+C Development and Cooperation]] |volume=49 |pages=285–289 |archive-url=https://web.archive.org/web/20081202113453/http://www.inwent.org/ez/articles/077943/index.en.shtml |archive-date=December 2, 2008 |access-date=July 18, 2020}}&lt;/ref&gt;&lt;ref&gt;{{cite report |date=May 2020 |title=The Philippines Corruption Report |url=https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-url=https://web.archive.org/web/20220812233543/https://www.ganintegrity.com/portal/country-profiles/the-philippines/ |archive-date=August 12, 2022 |access-date=August 7, 2020 |website=GAN Integrity}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last=Batalla |first=Eric V. C. |date=June 10, 2020 |title=Grand corruption scandals in the Philippines |journal=Public Administration and Policy |publisher=[[Emerald Group Publishing|Emerald Publishing Limited]] |volume=23 |issue=1 |pages=73–86 |doi=10.1108/PAP-11-2019-0036 |issn=2517-679X |doi-access=free}}&lt;/ref&gt; attributed by some historians to the Spanish colonial period's padrino system.&lt;ref&gt;{{cite book |editor-last1=Sriwarakuel |editor-first1=Warayuth |title=Cultural Traditions and Contemporary Challenges in Southeast Asia: Hindu and Buddhist |series=Cultural Heritage and Contemporary Change. Series IIID, South East Asia |volume=3 |date=2005 |publisher=Council for Research in Values and Philosophy |location=Washington, D.C. |isbn=978-1-56518-213-4 |page=[https://books.google.com/books?id=BnxpmvgAwcQC&amp;pg=PA294 294] |url=https://books.google.com/books?id=BnxpmvgAwcQC |editor-last2=Dy |editor-first2=Manuel B. |editor-last3=Haryatmoko |editor-first3=J. |editor-last4=Chuan |editor-first4=Nguyen Trong |editor-last5=Yiheang |editor-first5=Chhay |language=en |access-date=March 18, 2023 |archive-date=March 18, 2023 |archive-url=https://web.archive.org/web/20230318171623/https://books.google.com/books?id=BnxpmvgAwcQC |url-status=live}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last=Quah |first=Jon S. T. |url=https://books.google.com/books?id=7qV6un8vKNUC |title=Curbing Corruption in Asian Countries: An Impossible Dream? |series=Research in Public Policy Analysis and Management |volume=20 |date=2011 |publisher=[[Emerald Group Publishing]] |location=Bingley, West Yorkshire, England |isbn=978-0-85724-820-6 |pages=[https://books.google.com/books?id=7qV6un8vKNUC&amp;pg=115 115–117] |access-date=September 28, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074650/https://books.google.com/books?id=7qV6un8vKNUC |url-status=live}}&lt;/ref&gt; The Roman Catholic church exerts considerable but waning&lt;ref&gt;{{#invoke:cite|news|last1=Strother |first1=Jason |title=Power of the Catholic Church slipping in Philippines |url=https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |access-date=July 25, 2023 |work=[[The Christian Science Monitor]] |date=March 6, 2013 |archive-url=https://web.archive.org/web/20130307075323/https://www.csmonitor.com/World/Asia-Pacific/2013/0306/Power-of-the-Catholic-Church-slipping-in-Philippines |archive-date=March 7, 2013}}&lt;/ref&gt; influence in political affairs, although a constitutional provision for the separation of Church and State exists.&lt;ref&gt;{{cite journal |last1=Batalla |first1=Eric |last2=Baring |first2=Rito |title=Church-State Separation and Challenging Issues Concerning Religion |journal=[[Religions (journal)|Religions]] |date=March 15, 2019 |volume=10 |issue=3 |doi=10.3390/rel10030197 |at=Chapter 3: The Secular State and Church-State Separation, Chapter 4: Changing Church-State Relations |publisher=[[MDPI]] |language=en |issn=2077-1444 |doi-access=free}}&lt;/ref&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; 
'''''Executive branch'''''
The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt;

Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnicity'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace. 
Parliamentarians for Global Action (PGA)
Parliamentarians for Global Action (PGA)
 +4
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace. 
Parliamentarians for Global Action (PGA)
Parliamentarians for Global Action (PGA)
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace.
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace.
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
*Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia with the city of Manila as its capital. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace.
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
*Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia with the city of Manila as its capital. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

'''''Religion'''''
*Census data from 2020 found that 78.8 percent of the population professed Roman Catholicism.&lt;ref&gt;{{cite web |url=https://rsso05.psa.gov.ph/statistics/cph/node/1684057551|title=Religious Affiliation in Bicol Region (2020 Census of Population and Housing)|date=March 12, 2023|author=Republic of the Philippines|website=rsso05.psa.gov.ph|publisher=Philippine Statistics Authority|access-date=September 21, 2025 |url-status=live |archive-url=https://web.archive.org/web/20250921050435/https://rsso05.psa.gov.ph/statistics/cph/node/1684057551|archive-date=September 21, 2025 }}&lt;/ref&gt; Other Christian denominations include Iglesia ni Cristo, the Philippine Independent Church, and Seventh-day Adventism.&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot;&gt;{{Cite press release |last=Mapa |first=Dennis S. |author-link1=Dennis Mapa |date=February 22, 2023 |title=Religious Affiliation in the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/religious-affiliation-philippines-2020-census-population-and-housing |archive-url=https://web.archive.org/web/20230310184554/https://psa.gov.ph/content/religious-affiliation-philippines-2020-census-population-and-housing |archive-date=March 10, 2023 |access-date=March 12, 2023 |publisher=[[Philippine Statistics Authority]]}}&lt;/ref&gt; [[Protestantism in the Philippines|Protestants]] made up about 5% to 7% of the population in 2010.&lt;ref&gt;{{cite journal |last1=Jones |first1=Arun W. |title=Local Agency and the Reception of Protestantism in the Philippines |journal=Journal of Asian/North American Theological Educators |date=2016 |volume=2 |issue=1 |page=61 |url=http://janate.org/index.php/janate/article/download/1319/2198 |access-date=May 16, 2023 |archive-url=https://web.archive.org/web/20181109154652/http://janate.org/index.php/janate/article/download/1319/2198 |archive-date=November 9, 2018}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|web|title=Protestant Christianity in the Philippines |url=https://rlp.hds.harvard.edu/faq/protestant-christianity-philippines |archive-url=https://web.archive.org/web/20160421202509/https://rlp.hds.harvard.edu/faq/protestant-christianity-philippines |archive-date=April 21, 2016 |access-date=February 7, 2023 |website=Religious Literacy Project |publisher=[[Harvard Divinity School]]}}&lt;/ref&gt; The Philippines sends many [[Christian mission]]aries around the world, and is a training center for foreign priests and nuns.&lt;ref&gt;{{#invoke:cite|news|date=July 16, 2015 |title=Religious and lay Filipino missionaries in the world are &quot;Christ first witnesses |language=en |work=[[AsiaNews]] |url=https://www.asianews.it/news-en/Religious-and-lay-Filipino-missionariesin-the-world-are-%E2%80%9CChrist-first-witnesses-34790.html |access-date=April 23, 2022 |archive-url=https://web.archive.org/web/20220423154532/https://www.asianews.it/news-en/Religious-and-lay-Filipino-missionariesin-the-world-are-%E2%80%9CChrist-first-witnesses-34790.html |archive-date=April 23, 2022}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last1=Kim |first1=Sebastian |url=https://books.google.com/books?id=_YAdDQAAQBAJ |title=Christianity as a World Religion: An Introduction |edition=Second |last2=Kim |first2=Kirsteen |date=November 3, 2016 |publisher=[[Bloomsbury Publishing]] |location=London |isbn=978-1-4725-6936-3 |page=[https://books.google.com/books?id=_YAdDQAAQBAJ&amp;pg=PA70 70] |language=en |author-link1=Sebastian Kim |author-link2=Kirsteen Kim}}&lt;/ref&gt;

*Islam is the country's second-largest religion, with 6.4 percent of the population in the 2020 census.&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; Most Muslims live in Mindanao and nearby islands,&lt;ref name=&quot;StateGov-ReligiousFreedom-2015&quot; /&gt; and most adhere to the Shafi'i school of Sunni Islam.&lt;ref&gt;{{cite book |editor-last1=An-Na'im |editor-first1=Abdullahi |url=https://books.google.com/books?id=Hg0zCFM0fwkC |title=Islamic Family Law in a Changing World: A Global Resource Book |date=October 11, 2002 |publisher=[[Zed Books]] |location=London |isbn=978-1-84277-093-1 |page=[https://books.google.com/books?id=Hg0zCFM0fwkC&amp;pg=PA5 5]}}&lt;/ref&gt;

*About 0.2 percent of the population follow indigenous religions,&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; whose practices and folk beliefs are often syncretized with Christianity and Islam.&lt;ref name=&quot;Rodell-2002&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=y1CVR74_KHQC&amp;pg=PA29|name=29–30}}}}&lt;ref&gt;{{cite book |editor-last1=Min |editor-first1=Pyong Gap |url=https://books.google.com/books?id=EUx7AAAAQBAJ |title=Religions in Asian America: Building Faith Communities |editor-last2=Kim |editor-first2=Jung Ha |date=2001 |publisher=[[AltaMira Press]] |location=Walnut Creek, Calif. |isbn=978-1-4616-4762-1 |page=[https://books.google.com/books?id=EUx7AAAAQBAJ&amp;pg=PA144 144] |editor-link1=Pyong Gap Min}}&lt;/ref&gt; [[Buddhism in the Philippines|Buddhism]] is practiced by about 0.04% of the population,&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; primarily by Filipinos of Chinese descent.&lt;ref&gt;{{cite book |last=Yu |first=Jose Vidamor B. |url=https://books.google.com/books?id=c4WqAOKb5c8C |title=Inculturation of Filipino-Chinese Culture Mentality |series=Interreligious and Intercultural Investigations |volume=3 |date=2000 |publisher=[[Pontificia Università Gregoriana]] |location=Rome, Italy |isbn=978-88-7652-848-4 |pages=[https://books.google.com/books?id=c4WqAOKb5c8C&amp;pg=PA87 87–88]}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace.
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
*Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia with the city of Manila as its capital. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

'''''Religion'''''
*Census data from 2020 found that 78.8 percent of the population professed Roman Catholicism.&lt;ref&gt;{{cite web |url=https://rsso05.psa.gov.ph/statistics/cph/node/1684057551|title=Religious Affiliation in Bicol Region (2020 Census of Population and Housing)|date=March 12, 2023|author=Republic of the Philippines|website=rsso05.psa.gov.ph|publisher=Philippine Statistics Authority|access-date=September 21, 2025 |url-status=live |archive-url=https://web.archive.org/web/20250921050435/https://rsso05.psa.gov.ph/statistics/cph/node/1684057551|archive-date=September 21, 2025 }}&lt;/ref&gt; Other Christian denominations include Iglesia ni Cristo, the Philippine Independent Church, and Seventh-day Adventism.&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot;&gt;{{Cite press release |last=Mapa |first=Dennis S. |author-link1=Dennis Mapa |date=February 22, 2023 |title=Religious Affiliation in the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/religious-affiliation-philippines-2020-census-population-and-housing |archive-url=https://web.archive.org/web/20230310184554/https://psa.gov.ph/content/religious-affiliation-philippines-2020-census-population-and-housing |archive-date=March 10, 2023 |access-date=March 12, 2023 |publisher=[[Philippine Statistics Authority]]}}&lt;/ref&gt; Protestants made up about 5% to 7% of the population in 2010.&lt;ref&gt;{{cite journal |last1=Jones |first1=Arun W. |title=Local Agency and the Reception of Protestantism in the Philippines |journal=Journal of Asian/North American Theological Educators |date=2016 |volume=2 |issue=1 |page=61 |url=http://janate.org/index.php/janate/article/download/1319/2198 |access-date=May 16, 2023 |archive-url=https://web.archive.org/web/20181109154652/http://janate.org/index.php/janate/article/download/1319/2198 |archive-date=November 9, 2018}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|web|title=Protestant Christianity in the Philippines |url=https://rlp.hds.harvard.edu/faq/protestant-christianity-philippines |archive-url=https://web.archive.org/web/20160421202509/https://rlp.hds.harvard.edu/faq/protestant-christianity-philippines |archive-date=April 21, 2016 |access-date=February 7, 2023 |website=Religious Literacy Project |publisher=[[Harvard Divinity School]]}}&lt;/ref&gt; The Philippines sends many [[Christian mission]]aries around the world, and is a training center for foreign priests and nuns.&lt;ref&gt;{{#invoke:cite|news|date=July 16, 2015 |title=Religious and lay Filipino missionaries in the world are &quot;Christ first witnesses |language=en |work=[[AsiaNews]] |url=https://www.asianews.it/news-en/Religious-and-lay-Filipino-missionariesin-the-world-are-%E2%80%9CChrist-first-witnesses-34790.html |access-date=April 23, 2022 |archive-url=https://web.archive.org/web/20220423154532/https://www.asianews.it/news-en/Religious-and-lay-Filipino-missionariesin-the-world-are-%E2%80%9CChrist-first-witnesses-34790.html |archive-date=April 23, 2022}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last1=Kim |first1=Sebastian |url=https://books.google.com/books?id=_YAdDQAAQBAJ |title=Christianity as a World Religion: An Introduction |edition=Second |last2=Kim |first2=Kirsteen |date=November 3, 2016 |publisher=[[Bloomsbury Publishing]] |location=London |isbn=978-1-4725-6936-3 |page=[https://books.google.com/books?id=_YAdDQAAQBAJ&amp;pg=PA70 70] |language=en |author-link1=Sebastian Kim |author-link2=Kirsteen Kim}}&lt;/ref&gt;

*Islam is the country's second-largest religion, with 6.4 percent of the population in the 2020 census.&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; Most Muslims live in Mindanao and nearby islands,&lt;ref name=&quot;StateGov-ReligiousFreedom-2015&quot; /&gt; and most adhere to the Shafi'i school of Sunni Islam.&lt;ref&gt;{{cite book |editor-last1=An-Na'im |editor-first1=Abdullahi |url=https://books.google.com/books?id=Hg0zCFM0fwkC |title=Islamic Family Law in a Changing World: A Global Resource Book |date=October 11, 2002 |publisher=[[Zed Books]] |location=London |isbn=978-1-84277-093-1 |page=[https://books.google.com/books?id=Hg0zCFM0fwkC&amp;pg=PA5 5]}}&lt;/ref&gt;

*About 0.2 percent of the population follow indigenous religions,&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; whose practices and folk beliefs are often syncretized with Christianity and Islam.&lt;ref name=&quot;Rodell-2002&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=y1CVR74_KHQC&amp;pg=PA29|name=29–30}}}}&lt;ref&gt;{{cite book |editor-last1=Min |editor-first1=Pyong Gap |url=https://books.google.com/books?id=EUx7AAAAQBAJ |title=Religions in Asian America: Building Faith Communities |editor-last2=Kim |editor-first2=Jung Ha |date=2001 |publisher=[[AltaMira Press]] |location=Walnut Creek, Calif. |isbn=978-1-4616-4762-1 |page=[https://books.google.com/books?id=EUx7AAAAQBAJ&amp;pg=PA144 144] |editor-link1=Pyong Gap Min}}&lt;/ref&gt; Buddhism is practiced by about 0.04% of the population,&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; primarily by Filipinos of Chinese descent.&lt;ref&gt;{{cite book |last=Yu |first=Jose Vidamor B. |url=https://books.google.com/books?id=c4WqAOKb5c8C |title=Inculturation of Filipino-Chinese Culture Mentality |series=Interreligious and Intercultural Investigations |volume=3 |date=2000 |publisher=[[Pontificia Università Gregoriana]] |location=Rome, Italy |isbn=978-88-7652-848-4 |pages=[https://books.google.com/books?id=c4WqAOKb5c8C&amp;pg=PA87 87–88]}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace.
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
*Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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{{Comparative law and justice project|region=Asia}}[[User:CarlessParking|CarlessParking]] ([[User talk:CarlessParking|discuss]] • [[Special:Contributions/CarlessParking|contribs]]) 12:44, 11 April 2026 (UTC)

==Basic information==
[[File:Flag of the Philippines.svg|thumb|right|Flag of the Philippines]]
[[File:Philippines relief location map (square).svg|thumb|right|Map of the Philippines]]
'''''Geographic information'''''
*The Philippines is an archipelagic country in Southeast Asia with the city of Manila as its capital. In the western Pacific Ocean, it consists of 7,641 islands which are broadly categorized in three main geographical divisions from north to south: Luzon, Visayas, and Mindanao. The Philippines is bounded by the South China Sea to the west, the Philippine Sea to the east, and the Celebes Sea to the south. It shares maritime borders with Taiwan to the north, Japan to the northeast, Palau to the east and southeast, Indonesia to the south, Malaysia to the southwest, Vietnam to the west, and China to the northwest.

'''''Population'''''
*In July 2024, the Philippines had a population of 112,729,484.&lt;ref name=&quot;PSA_POPCEN&quot;/&gt; More than 60 percent of the country's population live in the coastal zone.&lt;ref&gt;{{cite book |author1=Department of Environment and Natural Resources |author2=Bureau of Fisheries and Aquatic Resources of the Department of Agriculture |author3=Department of the Interior and Local Government |author1-link=Department of Environment and Natural Resources |author2-link=Bureau of Fisheries and Aquatic Resources |author3-link=Department of the Interior and Local Government |title=Coastal Management Orientation and Overview |series=Philippine Coastal Management Guidebook Series |volume=1 |url=https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |via=Foreign-Assisted and Special Projects Service (FASPS) eLibrary |publisher=Coastal Resource Management Project of the [[Department of Environment and Natural Resources]] |access-date=August 7, 2024 |location=Cebu City, Philippines |date=2001 |isbn=978-971-92289-0-5 |pages=4, 9 |archive-url=https://web.archive.org/web/20180712173748/https://faspselib.denr.gov.ph/sites/default/files/Publication%20Files/crmguidebook1.pdf |archive-date=July 12, 2018}}&lt;/ref&gt; In 2020, 54 percent lived in urban areas.&lt;ref name=&quot;PSAGovPH-2020-Urban&quot;&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Urban Population of the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/urban-population-philippines-2020-census-population-and-housing |publisher=[[Philippine Statistics Authority]] |access-date=May 20, 2023 |archive-url=https://web.archive.org/web/20220705104809/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_PR_Urban%20Population_RML_063022_ONS-signed.pdf |archive-date=July 5, 2022 |date=July 4, 2022}}&lt;/ref&gt; Manila, its capital, and Quezon City, the country's most populous city, are in Metro Manila. About 13.48&amp;nbsp;million people, {{#expr: (13484462/109033245)*100 round 0}} percent of the Philippines' population, live in Metro Manila,&lt;ref name=&quot;PSAGovPH-2020-Urban&quot; /&gt; the country's most populous metropolitan area&lt;ref&gt;{{cite report |year=2017 |title=Philippine Development Plan, 2017–2022 |url=https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |chapter=Chapter 3: Overlay of Economic Growth, Demographic Trends, and Physical Characteristics |chapter-url=http://pdp.neda.gov.ph/wp-content/uploads/2017/01/Chapter-03.pdf |publisher=[[National Economic and Development Authority]] |location=Pasig, Philippines |issn=2243-7576 |pages=31, 34–35 |archive-url=https://web.archive.org/web/20200226035525/https://pdp.neda.gov.ph/wp-content/uploads/2017/01/PDP-2017-2022-10-03-2017.pdf |archive-date=February 26, 2020 |access-date=April 23, 2023}}&lt;/ref&gt; and the world's fifth most populous.&lt;ref&gt;{{cite report |url=http://www.demographia.com/db-worldua.pdf |title=Demographia World Urban Areas |date=July 2022 |publisher=[[Demographia]] |edition=18th Annual |page=23 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20230203065121/http://www.demographia.com/db-worldua.pdf |archive-date=February 3, 2023}}&lt;/ref&gt; Between 1948 and 2010, the population of the Philippines increased almost fivefold, from 19&amp;nbsp;million to 92&amp;nbsp;million.&lt;ref&gt;{{#invoke:cite|web|title=Population of the Philippines: Census Years 1799 to 2010 |url=http://www.nscb.gov.ph/secstat/d_popn.asp |archive-url=https://web.archive.org/web/20120704171010/http://www.nscb.gov.ph/secstat/d_popn.asp |archive-date=July 4, 2012 |website=[[Philippine Statistics Authority|National Statistical Coordination Board]] |access-date=July 24, 2023}}&lt;/ref&gt;

*The country's median age is 25.3, and 63.9 percent of its population is between 15 and 64 years old.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Age and Sex Distribution in the Philippine Population (2020 Census of Population and Housing) |url=https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |access-date=May 19, 2023 |work=[[Philippine Statistics Authority]] |date=August 12, 2022 |archive-url=https://web.archive.org/web/20220812232228/https://psa.gov.ph/sites/default/files/attachments/hsd/pressrelease/1_Press%20Release%20on_Age%20Sex_RML_18July22_rev_mpe_RRDH_CRD-signed.pdf |archive-date=August 12, 2022}}&lt;/ref&gt; The Philippines' average annual population growth rate is decreasing,&lt;ref&gt;{{cite journal |date=June 2018 |title=2015 Census of Population |url=http://www.psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |journal=Census Facts and Figures |location=Quezon City, Philippines |publisher=[[Philippine Statistics Authority]] |page=11 |issn=0117-1453 |archive-url=https://web.archive.org/web/20220814193607/https://psa.gov.ph/sites/default/files/_2015_Census%20Facts%20and%20Figures_Philippines_MERGE.pdf |archive-date=August 14, 2022 |access-date=July 25, 2020}}&lt;/ref&gt; although government attempts to further reduce population growth have been contentious.&lt;ref&gt;{{#invoke:cite|news|date=September 29, 2010 |title=Bishops threaten civil disobedience over RH bill |work=[[GMA News Online|GMANews.TV]] |url=http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |access-date=October 16, 2010 |archive-url=https://web.archive.org/web/20110221140718/http://www.gmanews.tv/100days/story/202186/bishops-threaten-civil-disobedience-over-rh-bill |archive-date=February 21, 2011}}&lt;/ref&gt; The country reduced its poverty rate from 49.2 percent in 1985,&lt;ref name=&quot;WorldBank-Poverty-2023&quot;&gt;{{cite report |title=Overcoming Poverty and Inequality in the Philippines; Past, Present, and Prospects for the Future |url=https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |publisher=[[The World Bank]] |access-date=April 2, 2023 |archive-url=https://web.archive.org/web/20221206083125/https://documents1.worldbank.org/curated/en/099325011232224571/pdf/P17486101e29310810abaf0e8e336aed85a.pdf |archive-date=December 6, 2022 |page=3}}&lt;/ref&gt; to 18.1 percent in 2021.&lt;ref&gt;{{Cite press release |last1=Mapa |first1=Claire Dennis S. |author-link1=Dennis Mapa |date=August 15, 2022 |title=Proportion of Poor Filipinos was Recorded at 18.1 Percent in 2021 |work=[[Philippine Statistics Authority]] |url=https://psa.gov.ph/poverty-press-releases/nid/167972 |access-date=November 8, 2022 |archive-url=https://web.archive.org/web/20220816035933/https://psa.gov.ph/poverty-press-releases/nid/167972 |archive-date=August 16, 2022}}&lt;/ref&gt; The Philippines' income inequality began to decline in 2012.&lt;ref name=&quot;WorldBank-Poverty-2023&quot; /&gt;

'''''Ethnic Groups'''''
*The Philippines has substantial ethnic diversity, due to foreign influence and the archipelago's division by water and topography.&lt;ref name=&quot;Banlaoi-2009&quot;&gt;{{cite book |last=Banlaoi |first=Rommel |url=https://books.google.com/books?id=hi_NBQAAQBAJ |title=Philippine Security in the Age of Terror: National, Regional, and Global Challenges in the Post-9/11 World |date=2009 |publisher=[[CRC Press]] |location=Boca Raton, Fla. |isbn=978-1-4398-1551-9 |pages=[https://books.google.com/books?id=hi_NBQAAQBAJ&amp;pg=PA31 31–32] |author-link1=Rommel Banlaoi}}&lt;/ref&gt; In the 2020 census, the Philippines' largest ethnic groups were Tagalog (26.0 percent), Visayans [excluding the Cebuano, Hiligaynon, and Waray] (14.3 percent), Ilocano and Cebuano (both eight percent), Hiligaynon (7.9 percent), Bicolano (6.5 percent), and Waray (3.8 percent).&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; The country's indigenous peoples consisted of 110 enthnolinguistic groups,&lt;ref&gt;{{cite report |date=February 2010 |title=Fast Facts: Indigenous Peoples in the Philippines |url=https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-url=https://web.archive.org/web/20230225201804/https://www.undp.org/sites/g/files/zskgke326/files/migration/ph/fastFacts6---Indigenous-Peoples-in-the-Philippines-rev-1.5.pdf |archive-date=February 25, 2023 |access-date=February 25, 2023 |publisher=[[United Nations Development Programme]]}}&lt;/ref&gt; with a combined population of 15.56&amp;nbsp;million, in 2020.&lt;ref name=&quot;PSAGovPH-Ethnicity-2020Census&quot; /&gt; They include the Igorot, Lumad, Mangyan, and indigenous peoples of Palawan.&lt;ref&gt;{{Cite tech report |last=Cariño |first=Jacqueline K. |date=November 2012 |title=Country Technical Note on Indigenous Peoples' Issues; Republic of the Philippines |url=https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-url=https://web.archive.org/web/20210809025044/https://www.ifad.org/documents/38714170/40224860/philippines_ctn.pdf/ae0faa4a-2b65-4026-8d42-219db776c50d |archive-date=August 9, 2021 |access-date=November 11, 2022 |publisher=[[International Fund for Agricultural Development]] |pages=3–5, 31–47}}&lt;/ref&gt;

*Negritos are thought to be among the islands' earliest inhabitants.&lt;ref name=&quot;Dolan-1991&quot; /&gt;{{rp|loc={{plain link|url=https://web.archive.org/web/20060927160916/http://countrystudies.us/philippines/35.htm|name=35}}}} These minority aboriginal settlers are an Australoid group, a remnant of the first human migration from Africa to Australia who were probably displaced by later waves of migration.&lt;ref&gt;{{cite book |last=Flannery |first=Tim |url=https://books.google.com/books?id=eIW5aktgo0IC |title=The Future Eaters: An Ecological History of the Australasian Lands and People |date=2002 |publisher=[[Grove Press]] |location=New York |isbn=978-0-8021-3943-6 |page=[https://books.google.com/books?id=eIW5aktgo0IC&amp;pg=PA147 147] |author-link1=Tim Flannery}}&lt;/ref&gt; Some Philippine Negritos have a Denisovan admixture in their genome.&lt;ref&gt;{{#invoke:cite|news|date=August 31, 2012 |title=Extinct humanoid species may have lived in PHL |language=en |work=[[GMA News Online]] |url=https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20181227040611/https://www.gmanetwork.com/news/scitech/science/272046/extinct-humanoid-species-may-have-lived-in-phl/story/ |archive-date=December 27, 2018}}&lt;/ref&gt;&lt;ref&gt;{{cite journal |last1=Reich |first1=David |last2=Patterson |first2=Nick |last3=Kircher |first3=Martin |last4=Delfin |first4=Frederick |last5=Nandineni |first5=Madhusudan R. |last6=Pugach |first6=Irina |last7=Ko |first7=Albert Min-Shan |last8=Ko |first8=Ying-Chin |last9=Jinam |first9=Timothy A. |last10=Phipps |first10=Maude E. |last11=Saitou |first11=Naruya |last12=Wollstein |first12=Andreas |last13=Kayser |first13=Manfred |last14=Pääbo |first14=Svante |last15=Stoneking |first15=Mark |date=October 2011 |title=Denisova Admixture and the First Modern Human Dispersals into Southeast Asia and Oceania |journal=[[American Journal of Human Genetics]] |volume=89 |issue=4 |pages=516–528 |doi=10.1016/j.ajhg.2011.09.005 |pmc=3188841 |pmid=21944045 |doi-access=free}}&lt;/ref&gt; Ethnic Filipinos generally belong to several Southeast Asian ethnic groups, classified linguistically as Austronesians speaking Malayo-Polynesian languages.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt;

'''''Languages'''''
*Filipino and English are the country's official languages.&lt;ref name=&quot;GovPH-OfficialLanguage&quot; /&gt; Filipino, a standardized version of Tagalog, is spoken primarily in Metro Manila.&lt;ref&gt;{{cite book |last=Takacs |first=Sarolta |url=https://books.google.com/books?id=u1TrBgAAQBAJ |title=The Modern World: Civilizations of Africa, Civilizations of Europe, Civilizations of the Americas, Civilizations of the Middle East and Southwest Asia, Civilizations of Asia and the Pacific |date=2015 |publisher=[[Routledge]] |location=London |isbn=978-1-317-45572-1 |page=[https://books.google.com/books?id=u1TrBgAAQBAJ&amp;pg=PA659 659]}}&lt;/ref&gt; Filipino and English are used in government, education, print, broadcast media, and business, often with a third local language.&lt;ref name=&quot;Brown-Ganguly-2003&quot;&gt;{{cite book |editor-last1=Brown |editor-first1=Michael Edward |url=https://books.google.com/books?id=fcoDezu1ABoC |title=Fighting Words: Language Policy and Ethnic Relations in Asia |series=BCSIA Studies in International Security |editor-last2=Ganguly |editor-first2=Sumit |date=2003 |publisher=[[MIT Press]] |location=Cambridge, Mass. |isbn=978-0-262-52333-2 |pages=[https://books.google.com/books?id=fcoDezu1ABoC&amp;pg=PA323 323–325] |editor-link2=Sumit Ganguly}}&lt;/ref&gt; Code-switching between English and other local languages, notably Tagalog, is common.&lt;ref&gt;{{cite journal |last1=Bautista |first1=Maria Lourdes S. |title=Tagalog-English Code Switching as a Mode of Discourse |journal=Asia Pacific Education Review |date=June 2004 |volume=5 |issue=2 |pages=226–231 |doi=10.1007/BF03024960 |url=https://files.eric.ed.gov/fulltext/EJ720543.pdf |access-date=July 3, 2023 |publisher=Education Research Institute, [[Seoul National University]] |issn=1598-1037 |oclc=425894528 |s2cid=145684166}}&lt;/ref&gt; The Philippine constitution provides for Spanish and Arabic on a voluntary, optional basis.&lt;ref name=&quot;GovPH-OfficialLanguage&quot;&gt;{{Cite constitution |article=XIV |section=7 |polity=the Philippines |date=1987 |url=https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-url=https://web.archive.org/web/20170609073807/https://www.officialgazette.gov.ph/constitutions/the-1987-constitution-of-the-republic-of-the-philippines/the-1987-constitution-of-the-republic-of-the-philippines-article-xiv/ |archive-date=June 9, 2017 |access-date=February 11, 2023 |website=[[Official Gazette of the Republic of the Philippines]]}}&lt;/ref&gt; Spanish, a widely used lingua franca during the late nineteenth century, has declined greatly in use,&lt;ref&gt;{{cite book |last=Stewart |first=Miranda |url=https://books.google.com/books?id=tfaUqzf1ht8C |title=The Spanish Language Today |date=2012 |publisher=[[Routledge]] |location=London |isbn=978-1-134-76548-5 |page=[https://books.google.com/books?id=tfaUqzf1ht8C&amp;pg=PA9 9]}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|news|last=Weedon |first=Alan |date=August 10, 2019 |title=The Philippines is fronting up to its Spanish heritage, and for some it's paying off |work=[[ABC News (Australia)|ABC News]] |publisher=[[Australian Broadcasting Corporation]] |url=https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-url=https://web.archive.org/web/20190810044706/https://www.abc.net.au/news/2019-08-10/inside-the-push-to-bring-back-spanish-into-the-philippines/11356590 |archive-date=August 10, 2019 |access-date=March 31, 2023}}&lt;/ref&gt; although Spanish loanwords are still present in Philippine languages.&lt;ref&gt;{{cite book |type=Conference proceeding |url=https://books.google.com/books?id=wG08AAAAIAAJ |title=Pidginization and Creolization of Languages; Proceedings of a Conference Held at the University of the West Indies, Mona, Jamaica, April 1968 |date=1971 |publisher=[[Cambridge University Press]] |location=Cambridge, England |isbn=978-0-521-09888-5 |editor-last=Hymes |editor-first=Dell |page=[https://books.google.com/books?id=wG08AAAAIAAJ&amp;pg=PA223 223] |author-link1=Dell Hymes}}&lt;/ref&gt;&lt;ref name=&quot;:2&quot;&gt;{{cite book |last1=Aspillera |first1=Paraluman S. |url=https://books.google.com/books?id=y8bZAwAAQBAJ |title=Basic Tagalog for Foreigners and Non-Tagalogs (with Online Audio) |edition=Revised Third |last2=Hernandez |first2=Yolanda Canseco |date=July 1, 2014 |publisher=[[Tuttle Publishing]] |location=North Clarendon, Vt. |isbn=978-1-4629-0166-1 |page=[https://books.google.com/books?id=y8bZAwAAQBAJ&amp;pg=PT10 9] |language=en}}&lt;/ref&gt;&lt;ref&gt;{{cite book |editor-last1=Allan |editor-first1=Keith |url=https://books.google.com/books?id=Omn6DwAAQBAJ |title=Dynamics of Language Changes: Looking Within and Across Languages |date=August 31, 2020 |publisher=[[Springer Nature]] |location=Singapore |isbn=978-981-15-6430-7 |page=[https://books.google.com/books?id=Omn6DwAAQBAJ&amp;pg=PA204 204] |language=en}}&lt;/ref&gt; Arabic is primarily taught in Mindanao Islamic schools.&lt;ref&gt;{{cite journal |last1=Samid |first1=Amina |title=Islamic Education and the Development of Madrasah Schools in the Philippines |journal=International Journal of Political Studies |date=August 31, 2022 |volume=8 |issue=2 |pages=37, 41–44 |doi=10.25272/icps.1139650 |url=https://dergipark.org.tr/en/download/article-file/2521857 |access-date=August 14, 2024 |issn=2528-9969 |via=DergiPark Akademik}}&lt;/ref&gt;

*In 2020, the top languages generally spoken at home were Tagalog, Binisaya, Hiligaynon, Ilocano, Cebuano, and Bikol.&lt;ref&gt;{{cite press release |last1=Mapa |first1=Dennis S. |author-link1=Dennis Mapa |title=Tagalog is the Most Widely Spoken Language at Home (2020 Census of Population and Housing) |url=https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |website=[[Philippine Statistics Authority]] |access-date=January 8, 2024 |archive-url=https://web.archive.org/web/20240108195246/https://psa.gov.ph/statistics/population-and-housing/node/1684041577 |archive-date=January 8, 2024 |date=March 7, 2023}}&lt;/ref&gt; Nineteen regional languages are auxiliary official languages as media of instruction:&lt;ref name=&quot;GMA-DepEd-7-Languages&quot;&gt;{{#invoke:cite|news|date=July 13, 2013 |title=DepEd adds 7 languages to mother tongue-based education for Kinder to Grade 3 |language=en |work=[[GMA News Online]] |url=http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |access-date=February 8, 2023 |archive-url=https://web.archive.org/web/20131216045522/http://www.gmanetwork.com/news/story/317280/news/nation/deped-adds-7-languages-to-mother-tongue-based-education-for-kinder-to-grade-3 |archive-date=December 16, 2013}}&lt;/ref&gt;

::* Aklanon
::* Bikol
::* Cebuano
::* Chavacano
::* Hiligaynon
::* Ibanag
::* Ilocano
::* Ivatan
::* Kapampangan
::* Kinaray-a
::* Maguindanao
::* Maranao
::* Pangasinan
::* Sambal
::* Surigaonon
::* Tagalog
::* Tausug
::* Waray
::* Yakan

Other indigenous languages, including Cuyonon, Ifugao, Itbayat, Kalinga, Kamayo, Kankanaey, Masbateño, Romblomanon, Manobo, and several Visayan languages, are used in their respective provinces.&lt;ref name=&quot;Ethnologue-PH&quot; /&gt; Filipino Sign Language is the national sign language, and the language of deaf education.&lt;ref&gt;{{#invoke:cite|news|last=Kabiling |first=Genalyn |date=November 12, 2018 |title=Filipino Sign Language declared as nat'l sign language of Filipino deaf |work=[[Manila Bulletin]] |url=https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |access-date=November 12, 2018 |archive-url=https://web.archive.org/web/20181112122321/https://news.mb.com.ph/2018/11/12/filipino-sign-language-declared-as-natl-sign-language-of-filipino-deaf/ |archive-date=November 12, 2018}}&lt;/ref&gt;

'''''Religion'''''
*Census data from 2020 found that 78.8 percent of the population professed Roman Catholicism.&lt;ref&gt;{{cite web |url=https://rsso05.psa.gov.ph/statistics/cph/node/1684057551|title=Religious Affiliation in Bicol Region (2020 Census of Population and Housing)|date=March 12, 2023|author=Republic of the Philippines|website=rsso05.psa.gov.ph|publisher=Philippine Statistics Authority|access-date=September 21, 2025 |url-status=live |archive-url=https://web.archive.org/web/20250921050435/https://rsso05.psa.gov.ph/statistics/cph/node/1684057551|archive-date=September 21, 2025 }}&lt;/ref&gt; Other Christian denominations include Iglesia ni Cristo, the Philippine Independent Church, and Seventh-day Adventism.&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot;&gt;{{Cite press release |last=Mapa |first=Dennis S. |author-link1=Dennis Mapa |date=February 22, 2023 |title=Religious Affiliation in the Philippines (2020 Census of Population and Housing) |url=https://psa.gov.ph/content/religious-affiliation-philippines-2020-census-population-and-housing |archive-url=https://web.archive.org/web/20230310184554/https://psa.gov.ph/content/religious-affiliation-philippines-2020-census-population-and-housing |archive-date=March 10, 2023 |access-date=March 12, 2023 |publisher=[[Philippine Statistics Authority]]}}&lt;/ref&gt; Protestants made up about 5% to 7% of the population in 2010.&lt;ref&gt;{{cite journal |last1=Jones |first1=Arun W. |title=Local Agency and the Reception of Protestantism in the Philippines |journal=Journal of Asian/North American Theological Educators |date=2016 |volume=2 |issue=1 |page=61 |url=http://janate.org/index.php/janate/article/download/1319/2198 |access-date=May 16, 2023 |archive-url=https://web.archive.org/web/20181109154652/http://janate.org/index.php/janate/article/download/1319/2198 |archive-date=November 9, 2018}}&lt;/ref&gt;&lt;ref&gt;{{#invoke:cite|web|title=Protestant Christianity in the Philippines |url=https://rlp.hds.harvard.edu/faq/protestant-christianity-philippines |archive-url=https://web.archive.org/web/20160421202509/https://rlp.hds.harvard.edu/faq/protestant-christianity-philippines |archive-date=April 21, 2016 |access-date=February 7, 2023 |website=Religious Literacy Project |publisher=[[Harvard Divinity School]]}}&lt;/ref&gt; The Philippines sends many Christian missionaries around the world, and is a training center for foreign priests and nuns.&lt;ref&gt;{{#invoke:cite|news|date=July 16, 2015 |title=Religious and lay Filipino missionaries in the world are &quot;Christ first witnesses |language=en |work=[[AsiaNews]] |url=https://www.asianews.it/news-en/Religious-and-lay-Filipino-missionariesin-the-world-are-%E2%80%9CChrist-first-witnesses-34790.html |access-date=April 23, 2022 |archive-url=https://web.archive.org/web/20220423154532/https://www.asianews.it/news-en/Religious-and-lay-Filipino-missionariesin-the-world-are-%E2%80%9CChrist-first-witnesses-34790.html |archive-date=April 23, 2022}}&lt;/ref&gt;&lt;ref&gt;{{cite book |last1=Kim |first1=Sebastian |url=https://books.google.com/books?id=_YAdDQAAQBAJ |title=Christianity as a World Religion: An Introduction |edition=Second |last2=Kim |first2=Kirsteen |date=November 3, 2016 |publisher=[[Bloomsbury Publishing]] |location=London |isbn=978-1-4725-6936-3 |page=[https://books.google.com/books?id=_YAdDQAAQBAJ&amp;pg=PA70 70] |language=en |author-link1=Sebastian Kim |author-link2=Kirsteen Kim}}&lt;/ref&gt;

*Islam is the country's second-largest religion, with 6.4 percent of the population in the 2020 census.&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; Most Muslims live in Mindanao and nearby islands,&lt;ref name=&quot;StateGov-ReligiousFreedom-2015&quot; /&gt; and most adhere to the Shafi'i school of Sunni Islam.&lt;ref&gt;{{cite book |editor-last1=An-Na'im |editor-first1=Abdullahi |url=https://books.google.com/books?id=Hg0zCFM0fwkC |title=Islamic Family Law in a Changing World: A Global Resource Book |date=October 11, 2002 |publisher=[[Zed Books]] |location=London |isbn=978-1-84277-093-1 |page=[https://books.google.com/books?id=Hg0zCFM0fwkC&amp;pg=PA5 5]}}&lt;/ref&gt;

*About 0.2 percent of the population follow indigenous religions,&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; whose practices and folk beliefs are often syncretized with Christianity and Islam.&lt;ref name=&quot;Rodell-2002&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=y1CVR74_KHQC&amp;pg=PA29|name=29–30}}}}&lt;ref&gt;{{cite book |editor-last1=Min |editor-first1=Pyong Gap |url=https://books.google.com/books?id=EUx7AAAAQBAJ |title=Religions in Asian America: Building Faith Communities |editor-last2=Kim |editor-first2=Jung Ha |date=2001 |publisher=[[AltaMira Press]] |location=Walnut Creek, Calif. |isbn=978-1-4616-4762-1 |page=[https://books.google.com/books?id=EUx7AAAAQBAJ&amp;pg=PA144 144] |editor-link1=Pyong Gap Min}}&lt;/ref&gt; Buddhism is practiced by about 0.04% of the population,&lt;ref name=&quot;PSAGovPH-2020Census-Religion&quot; /&gt; primarily by Filipinos of Chinese descent.&lt;ref&gt;{{cite book |last=Yu |first=Jose Vidamor B. |url=https://books.google.com/books?id=c4WqAOKb5c8C |title=Inculturation of Filipino-Chinese Culture Mentality |series=Interreligious and Intercultural Investigations |volume=3 |date=2000 |publisher=[[Pontificia Università Gregoriana]] |location=Rome, Italy |isbn=978-88-7652-848-4 |pages=[https://books.google.com/books?id=c4WqAOKb5c8C&amp;pg=PA87 87–88]}}&lt;/ref&gt;

==Brief history==
In 1896, the Philippine Revolution began. In December 1897, the Spanish government and the revolutionaries signed a truce, the Pact of Biak-na-Bato, requiring that the Spaniards pay the revolutionaries $MXN800,000 {{inflation|US|1|1897|r=2}}/2 round 2 today.&lt;ref&gt;{{cite book|last=Halstead|first=Murat|title=The Story of the Philippines and Our New Possessions|url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=1|chapter=XII. The American Army in Manila|chapter-url=http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=122|year=1898|publication-date=May 22, 2004|page=[http://www.gutenberg.org/catalog/world/readfile?fk_files=58428&amp;pageno=126 126].&lt;/ref&gt; The ''peso fuerte'' and the Mexican dollar were interchangeable at par. and that Aguinaldo and other leaders go into exile in Hong Kong. In April 1898, shortly after the beginning of the Spanish–American War, Commodore George Dewey, aboard the USS ''Olympia'', sailed into Manila Bay, leading the Asiatic Squadron of the US Navy. On May 1, 1898, the US defeated the Spaniards in the Battle of Manila Bay. Emilio Aguinaldo decided to return to the Philippines to help American forces defeat the Spaniards. The US Navy agreed to transport him back aboard the USS ''McCulloch'', and on May 19, he arrived in Cavite.&lt;ref&gt;Agoncillo, page 157&lt;/ref&gt;
[[File:NDS reverse 5 Philippine peso bill.jpg|thumb|The Proclamation of Independence on June 12, 1898, as depicted on the back of the Philippine five peso bill]]
[[File:Site of the Proclamation of Philippine Indepedence Historical Markers, Kawit, Cavite.jpg|thumb|left|Proclamation of Philippine Independence Historical Markers at the Aguinaldo Shrine in Kawit, Cavite]]

Independence was proclaimed on June 12, 1898, between four and five in the afternoon in Cavite at the ancestral home of General Emilio Aguinaldo in Cavite el Viejo (present-day Kawit), Cavite, some {{convert|30|km|sp=us}} south of Manila. The event saw the unfurling of the flag of the Philippines, made in Hong Kong by Marcela Agoncillo, Lorenza Agoncillo, and Delfina Herboza, and the performance of the ''Marcha Filipina Magdalo'', as the national anthem, now known as ''Lupang Hinirang'', which was composed by Julián Felipe and played by the ''San Francisco de Malabon'' marching band.

==Governance==
The Philippines has a democratic government, a constitutional republic with a presidential system.&lt;ref name=&quot;Rose-Ackerman&quot;&gt;{{cite journal |last1=Rose-Ackerman |first1=Susan |last2=Desierto |first2=Diane A. |last3=Volosin |first3=Natalia |date=2011 |title=Hyper-Presidentialism: Separation of Powers without Checks and Balances in Argentina and Philippines |url=https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |journal=[[Berkeley Journal of International Law]] |publisher=[[UC Berkeley School of Law]] |volume=29 |oclc=8092527577 |pages=246–333 |author-link1=Susan Rose-Ackerman |archive-url=https://web.archive.org/web/20220126072232/https://openyls.law.yale.edu/bitstream/handle/20.500.13051/3618/29BerkeleyJIntlL246.pdf?sequence=2&amp;isAllowed=y |archive-date=January 26, 2022}}&lt;/ref&gt;&lt;br&gt; 
'''''Executive branch'''''
*The president is head of state and head of government,&lt;ref name=&quot;Banlaoi-2009&quot;/&gt; and is the commander-in-chief of the armed forces.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt; The president is elected through direct election by the citizens of the Philippines for a six-year term.&lt;ref&gt;{{cite journal |last1=Teehankee |first1=Julio C. |author-link1=Julio C. Teehankee |last2=Thompson |first2=Mark R. |date=October 2016 |title=The Vote in the Philippines: Electing A Strongman |url=https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |journal=[[Journal of Democracy]] |publisher=[[Johns Hopkins University Press]] |volume=27 |issue=4 |issn=1086-3214 |pages=124–134 |doi=10.1353/jod.2016.0068 |author-link2=Mark R. Thompson |access-date=December 30, 2020 |archive-date=January 17, 2021 |archive-url=https://web.archive.org/web/20210117011258/https://www.journalofdemocracy.org/articles/the-vote-in-the-philippines-electing-a-strongman/ |url-status=live|url-access=subscription }}&lt;/ref&gt; The vice president, limited to two consecutive six-year terms, is elected separately from the president.&lt;ref name=&quot;Lazo&quot;&gt;{{cite book |last1=Lazo |first1=Ricardo S. |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=Rex Bookstore, Inc. |isbn=978-971-23-4546-3 |edition=2006 |url=https://books.google.com/books?id=fMszAErMRKYC}}&lt;/ref&gt; This means the president and vice president may be from different political parties. The president appoints and presides over the cabinet and officials of various national government agencies and institutions.&lt;ref name=&quot;Lazo-2009&quot;&gt;{{cite book |last=Lazo |first=Ricardo S. Jr. |url=https://books.google.com/books?id=fMszAErMRKYC |title=Philippine Governance and the 1987 Constitution |date=2009 |publisher=[[REX Book Store, Inc.]] |location=Manila, Philippines |isbn=978-971-23-4546-3 |edition=2006 |access-date=December 30, 2020 |archive-date=February 3, 2024 |archive-url=https://web.archive.org/web/20240203074324/https://books.google.com/books?id=fMszAErMRKYC |url-status=live}}&lt;/ref&gt;{{rp|page={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=213|name=213–214}}}} 
'''''Legislative branch'''''
*The bicameral Congress is composed of the Senate (the upper house, with members elected to a six-year term) and the House of Representatives (the lower house, with members elected to a three-year term).&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot;&gt;{{cite report |title=Carter Center Limited Mission to the May 2010 Elections in the Philippines Final Report |url=https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-url=https://web.archive.org/web/20120323212046/https://www.cartercenter.org/resources/pdfs/news/peace_publications/election_reports/philippines-may%202010-elections-finalrpt.pdf |archive-date=March 23, 2012 |publisher=[[The Carter Center]] |location=Atlanta, Ga. |oclc=733049273}}&lt;/ref&gt; Senators are elected at-large,&lt;ref name=&quot;CarterCenterOrg-2010-Elections&quot; /&gt; and representatives are elected from legislative districts and party lists.&lt;ref name=&quot;Lazo-2009&quot; /&gt;{{rp|pages={{plain link|url=https://books.google.com/books?id=fMszAErMRKYC&amp;pg=162|name=162–163}}}} 
'''''Judicial branch'''''
*Judicial authority is vested in the Supreme Court, composed of a chief justice and fourteen associate justices,&lt;ref&gt;{{cite book |date=March 2001 |editor-last=Pangalangan |editor-first=Raul C. |title=The Philippine Judicial System |url=https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |series=Asian Law Series |publisher=[[Institute of Developing Economies]] |location=Chiba, Japan |oclc=862953657 |pages=6, 39 |archive-url=https://web.archive.org/web/20210305185845/https://aboutphilippines.org/doc-pdf-ppt-etc/05_Philippine-Judicial-System.pdf |archive-date=March 5, 2021 |author-link1=Raul Pangalangan}}&lt;/ref&gt; who are appointed by the president from nominations submitted by the Judicial and Bar Council.&lt;ref name=&quot;Rose-Ackerman&quot; /&gt;

==Court and Criminal Law==
===Punishment===
The Philippines was the only country aside from the United States that used the electric chair, which had been introduced during the American colonial period. Until its first abolition in 1987, the country had reverted to using death by firing squad.

After the re-introduction of the death penalty in 1993, the country switched to lethal injection as its sole method of execution.

*Capital Punishment: Prohibited since 2006, with ongoing legislative debates on reintroduction for drug-related crimes.
*Afflictive Penalties:
**Reclusion Perpetua: 20 years and 1 day to 40 years imprisonment.
**Reclusion Temporal: 12 years and 1 day to 20 years.
**Prision Mayor: 6 years and 1 day to 12 years.
*Correctional Penalties:
**Prision Correccional: 6 months and 1 day to 6 years.
**Arresto Mayor: 1 month and 1 day to 6 months.
*Light Penalties:
**Arresto Menor: 1 to 30 days.
*Public Censure.
*Accessory Penalties: Includes perpetual/temporary disqualification from public office, civil interdiction, and bond to keep the peace.
*Legal Context
**Exceptions: Children under 15 are exempt from criminal liability but subjected to intervention programs.
**Torture: Prohibited under RA 9745, which penalizes physical or mental abuse by authorities.
*Rehabilitation: The system, while punitive, includes provisions for rehabilitation and correctional reform.
**Penalties for Crimes: Specific crimes like rebellion or murder have defined ranges (e.g., murder is punishable by reclusion perpetua to death, though death is not currently implemented)

==Rights==
===Family law===
In 1987, President Corazon Aquino enacted into law The Family Code of 1987, which was intended to supplant Book I of the Civil Code concerning persons and family relations. Work on the Family Code had begun as early as 1979, and it had been drafted by two successive committees, the first chaired by future Supreme Court Justice Flerida Ruth Pineda-Romero|Ruth Romero, and the second chaired by former Supreme Court Justice J.B.L. Reyes. The Civil Code needed amendment via the Family Code in order to alter certain provisions derived from foreign sources which had proven unsuitable to Filipino culture and to attune it to contemporary developments and trends.&lt;ref&gt;{{cite book |last= Sempio-Diy|first= Alicia|title= Handbook on the Family Code of the Philippines|year= 1988|publisher= Central Lawbook Publishing Co., Inc.|location= Quezon City}}&lt;/ref&gt;

The Family Code covers fields of significant public interest, especially the laws on marriage. The definition and requisites for marriage, along with the grounds for annulment, are found in the Family Code, as is the law on conjugal property relations, rules on establishing filiation, and the governing provisions on child support, parental authority, and adoption.

::*Marriage
::*Legal separation
::*Spousal rights and obligations
::*Marital property schemes
::*The Family
::*Paternity and filiation
::*Adoption
::*Maintenance (e.g. alimony, child support)
::*Parental authority
::*Emancipation and age of majority
::*Summary judicial proceedings in family law

===Adoption===
On June 10, 1975, President Ferdinand E. Marcos enacted Presidential Decree No. 603 also known as the Child and Youth Welfare Code.&lt;ref&gt;Child and Youth Welfare Code&lt;/ref&gt; In here, the old provisions regarding who were allowed to adopt were repealed by Article 27. The adopter still had to be in full possession of his civil rights before he could adopt but now, even those who already had natural children or natural children by legal fiction were allowed to adopt as long as they could provide support and care for all the children.&lt;ref&gt;{{Cite book|title=Not Bone of My Bone but Still My Own: A Treatise on the Philippine Adoption|last=Pangalangan|first=Elizabeth A|publisher=University of the Philippines College of Law|year=2013|location=Quezon City|pages=31}}&lt;/ref&gt; Among the new provisions added were those requiring the Department of Social Welfare and Development to make a case study of the adoptee, his natural parents, and the prospective adoptive parents before an adoption petition may be granted.&lt;ref&gt;Child and Youth Welfare Code, Article 33&lt;/ref&gt; A trial custody of at least six months is also required from the prospective adoptive parents so that their &quot;adjustment and emotional readiness for the legal union&quot; may be determined.&lt;ref&gt;Child and Youth Welfare Code, Article 35&lt;/ref&gt; Once the court is satisfied with the results of the trial custody and the report of the DSWD, a decree of adoption, which states the name that the adopted child will be known from then on, is issued.&lt;ref&gt;Child and Youth Welfare Code, Article 36&lt;/ref&gt;

==Works Cited==

[[Category:Philippines]]</text>
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|Source={{own|Young1lim}}
|Date=2026-04-11
|Author=Young W. Lim
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Source={{own|Young1lim}}
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|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Source={{own|Young1lim}}
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|Source={{own|Young1lim}}
|Date=2026-04-11
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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      <comment>Mfield moved page [[User talk:MaRayneS]] to [[User talk:Marisatri]]: Automatically moved page while renaming the user &quot;[[Special:CentralAuth/MaRayneS|MaRayneS]]&quot; to &quot;[[Special:CentralAuth/Marisatri|Marisatri]]&quot;</comment>
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    <title>Reiter Family (Habsburg Monarchy)</title>
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      <comment>Created page with &quot;{{Learning resource |topic=History |level=Advanced }} {| class=&quot;wikitable&quot; style=&quot;float:right; width:300px; margin-left:15px;&quot; ! colspan=&quot;2&quot; style=&quot;text-align:center; font-size:120%;&quot; |Reiter Family |- | colspan=&quot;2&quot; style=&quot;text-align:center;&quot; |[[File:Reiter_v_Ritterfels_reconstruction.png|205x205px]] |- !Origin |Tyrol, [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]] |- !Type |Service lineage |- !Estate |Lengberg (historical association) |- !Traditions |Military, admini...&quot;</comment>
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{| class=&quot;wikitable&quot; style=&quot;float:right; width:300px; margin-left:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:center; font-size:120%;&quot; |Reiter Family
|-
| colspan=&quot;2&quot; style=&quot;text-align:center;&quot; |[[File:Reiter_v_Ritterfels_reconstruction.png|205x205px]]
|-
!Origin
|Tyrol, [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]]
|-
!Type
|Service lineage
|-
!Estate
|Lengberg (historical association)
|-
!Traditions
|Military, administrative, educational service
|-
!Motto
|''Disciplina et Virtus''
(&quot;Discipline and Virtue&quot;)
|}
The '''Reiter''' family is associated with traditions of imperial service within the [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]], particularly in the fields of military, administrative, and educational activity. The name is historically connected with Tyrol, where individuals bearing the surname ''Reiter'' are recorded in positions of judicial, military, and territorial authority from the 16th century onward.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entries for Reiter family (Lienz, Klausen, Innsbruck).&lt;/ref&gt;

The lineage developed within a tradition defined by disciplined service, military formation, and structured education, later associated with Vienna as the imperial center and subsequently extended into the southern regions of the Monarchy.

== History ==

=== Tyrolean origins ===
The earliest foundations of the House of Reiter are associated with Tyrol, a region of strategic, military, and administrative importance within the Habsburg lands, forming a key corridor between the Alpine territories and northern Italy.

From the late 16th century onward, individuals bearing the name ''Reiter'' appear in positions of authority within the structures of local governance and territorial administration. Documented roles include ''Stadtrichter'' (city judges) in Lienz and Klausen, as well as a ''Stadthauptmann'' (city military commander) in Innsbruck.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'': Ruep Reiter (Stadt-Richter zu Lienz, 1580–1582, 1591–1593); Gabriel Reiter (Stadtrichter in Klausen, 1625); Michael Reiter (former Stadthauptmann in Innsbruck, c. 1650).&lt;/ref&gt;

These offices were integral to the functioning of the early modern state. The ''Stadtrichter'' exercised judicial authority and local administrative oversight, while the ''Stadthauptmann'' was responsible for urban defense, military organization, and coordination with regional command structures.

The recurrence of such appointments across different Tyrolean centers indicates the presence of the name within a service-oriented administrative and military stratum, in which advancement was based on discipline, reliability, and demonstrated competence within institutional frameworks.

This environment formed the early foundation of the lineage’s association with structured authority, military readiness, and civic responsibility.

=== Reiter von Ritterfels ===
A defining historical expression of the lineage is associated with '''Franz Reiter von Ritterfels''', recorded between 1692 and 1702 as ''Pflegsverwalter'' at Lengberg.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entry for Franz Reiter v. Ritterfels.&lt;/ref&gt;

This office represented a territorial administrative function combining executive governance, judicial authority, fiscal oversight, and supervision of regional order within a defined jurisdiction.

The ''Pflegsverwalter'' operated as a direct representative of territorial authority, responsible for maintaining legal order, administering estates, and overseeing local administration on behalf of higher governing structures.

The designation ''von Ritterfels'' reflects an association with territorial identity and administrative responsibility within the imperial system.

A coat of arms attributed to Franz Reiter von Ritterfels is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

In later family usage, the shortened form “von Reiter” emerged as a preserved expression of this heritage, retaining its association with service, discipline, and administrative function.

=== Viennese phase and continuation of imperial service tradition ===
During the mid to late 19th century, a branch of the Reiter family is associated with Vienna, reflecting its integration into the military and administrative structures of the Habsburg Monarchy.

A documented representative of this branch is '''Franz von Reiter''', an officer of the imperial army whose service is recorded in the ''Militär-Schematismus'' between 1850 and 1876.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850–1876.&lt;/ref&gt;

He entered service as an ''Unterleutnant'' in the 38th Infantry Regiment and advanced through the ranks to ''Oberleutnant'', later attaining the rank of ''Hauptmann 1. Classe'' in the 32nd Hungarian Infantry Regiment.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850, 1855, 1859, 1868; ''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1869–1876.&lt;/ref&gt;

His assignments included service in Lombardy, Lower Austria, Hungary, and the Adriatic regions, reflecting the geographical mobility characteristic of imperial officers.&lt;ref&gt;''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1871–1876.&lt;/ref&gt;

His son, '''Franz von Reiter''' (born 1878 in Vienna), served in the imperial army and subsequently relocated to the southern territories of the Monarchy, where he became established in the Ivanec region. By 1906, he was the proprietor of a mill known locally as ''Rajterov mlin''.&lt;ref&gt;Ivanec regional historical materials, Kukuljević–Rajterov mlin.&lt;/ref&gt;

=== Southern branch ===
In the regional context, the family name appears in its localized form as ''Franjo Rajter'', reflecting linguistic adaptation within a multilingual imperial environment while preserving continuity of lineage identity.

Franz von Reiter (Franjo Rajter) was the father of seven children, consisting of six daughters and one son, '''Franjo Rajter''' (born 1917), who continued the family line.

The latter attended the [[wikipedia:University_of_Zagreb|University of Zagreb]], where he completed studies in law and economics, reflecting the continued emphasis on higher education and professional formation within the lineage. In his later career, he served as a managing director within the national railway system of Yugoslavia ([[wikipedia:Yugoslav_Railways|Jugoslavenske Željeznice]]), maintaining the tradition of institutional responsibility characteristic of the House.

== Coat of arms ==
[[File:Reiter_v_Ritterfels_archival.jpg|right|thumb|220x220px|Archival depiction of the coat of arms attributed to Franz Reiter von Ritterfels (1692–1702), Tiroler Landesmuseen Ferdinandeum.]]
[[File:Reiter_v_Ritterfels_reconstruction.png|right|thumb|226x226px|Modern artistic reconstruction based on archival reference, for illustrative purposes only.]]
A coat of arms associated with the name ''Reiter von Ritterfels'' is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

The armorial entry attributes the coat of arms to '''Franz Reiter von Ritterfels''' in his capacity as ''Pflegsverwalter'' at Lengberg. The surviving depiction reflects the heraldic style of Tyrol in the late 17th century, including a shield and helmet with crest.

No formal blazon is recorded in the archival entry. The depiction suggests a quartered shield featuring a mounted rider and a lion, surmounted by a helmet with crest, consistent with heraldic conventions of the period.

== Tradition and identity ==
The House of Reiter is characterized by a tradition rooted in:

* military discipline
* structured education
* administrative and institutional service
* restraint in outward expression of status

Rather than defining itself through hereditary rank alone, the lineage expresses identity through continuity of conduct, responsibility, and adherence to a disciplined cultural framework.

This reflects the broader model of the imperial service class, in which authority and standing were closely linked to function, education, and professional conduct.

== Motto ==
''Disciplina et Virtus''  (&quot;Discipline and Virtue&quot;)

== References ==
&lt;references /&gt;</text>
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{| class=&quot;wikitable&quot; style=&quot;float:right; width:300px; margin-left:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:center; font-size:120%;&quot; |Reiter Family
|-
| colspan=&quot;2&quot; style=&quot;text-align:center;&quot; |[[File:Reiter_v_Ritterfels_reconstruction.png|205x205px]]
|-
!Origin
|Tyrol, [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]]
|-
!Type
|Service lineage
|-
!Estate
|Lengberg (historical association)
|-
!Traditions
|Military, administrative, educational service
|-
!Motto
|''Disciplina et Virtus''
(&quot;Discipline and Virtue&quot;)
|}
The '''Reiter''' family is associated with traditions of imperial service within the [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]], particularly in the fields of military, administrative, and educational activity. The name is historically connected with Tyrol, where individuals bearing the surname ''Reiter'' are recorded in positions of judicial, military, and territorial authority from the 16th century onward.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entries for Reiter family (Lienz, Klausen, Innsbruck).&lt;/ref&gt;

The lineage developed within a tradition defined by disciplined service, military formation, and structured education, later associated with Vienna as the imperial center and subsequently extended into the southern regions of the Monarchy.

== History ==

=== Tyrolean origins ===
The earliest origins of the Reiter family are associated with Tyrol, a region of considerable strategic, military, and administrative importance within the Habsburg lands, serving as a vital corridor between the Alpine territories and northern Italy.

From the late 16th century onward, individuals bearing the name ''Reiter'' appear in positions of authority within the structures of local governance and territorial administration. Documented roles include ''Stadtrichter'' (city judges) in Lienz and Klausen, as well as a ''Stadthauptmann'' (city military commander) in Innsbruck.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'': Ruep Reiter (Stadt-Richter zu Lienz, 1580–1582, 1591–1593); Gabriel Reiter (Stadtrichter in Klausen, 1625); Michael Reiter (former Stadthauptmann in Innsbruck, c. 1650).&lt;/ref&gt;

These offices were integral to the functioning of the early modern state. The ''Stadtrichter'' exercised judicial authority and local administrative oversight, while the ''Stadthauptmann'' was responsible for urban defense, military organization, and coordination with regional command structures.

The recurrence of such appointments across different Tyrolean centers indicates the presence of the name within a service-oriented administrative and military stratum, in which advancement was based on discipline, reliability, and demonstrated competence within institutional frameworks.

This environment formed the early foundation of the lineage’s association with structured authority, military readiness, and civic responsibility.

=== Reiter von Ritterfels ===
A defining historical expression of the lineage is associated with '''Franz Reiter von Ritterfels''', recorded between 1692 and 1702 as ''Pflegsverwalter'' at Lengberg.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entry for Franz Reiter v. Ritterfels.&lt;/ref&gt;

This office represented a territorial administrative function combining executive governance, judicial authority, fiscal oversight, and supervision of regional order within a defined jurisdiction.

The ''Pflegsverwalter'' operated as a direct representative of territorial authority, responsible for maintaining legal order, administering estates, and overseeing local administration on behalf of higher governing structures.

The designation ''von Ritterfels'' reflects an association with territorial identity and administrative responsibility within the imperial system.

A coat of arms attributed to Franz Reiter von Ritterfels is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

In later family usage, the shortened form “von Reiter” emerged as a preserved expression of this heritage, retaining its association with service, discipline, and administrative function.

=== Viennese phase and continuation of imperial service tradition ===
During the mid to late 19th century, a branch of the Reiter family is associated with Vienna, reflecting its integration into the military and administrative structures of the Habsburg Monarchy.

A documented representative of this branch is '''Franz von Reiter''', an officer of the imperial army whose service is recorded in the ''Militär-Schematismus'' between 1850 and 1876.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850–1876.&lt;/ref&gt;

He entered service as an ''Unterleutnant'' in the 38th Infantry Regiment and advanced through the ranks to ''Oberleutnant'', later attaining the rank of ''Hauptmann 1. Classe'' in the 32nd Hungarian Infantry Regiment.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850, 1855, 1859, 1868; ''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1869–1876.&lt;/ref&gt;

His assignments included service in Lombardy, Lower Austria, Hungary, and the Adriatic regions, reflecting the geographical mobility characteristic of imperial officers.&lt;ref&gt;''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1871–1876.&lt;/ref&gt;

His son, '''Franz von Reiter''' (born 1878 in Vienna), served in the imperial army and subsequently relocated to the southern territories of the Monarchy, where he became established in the Ivanec region. By 1906, he was the proprietor of a mill known locally as ''Rajterov mlin''.&lt;ref&gt;Ivanec regional historical materials, Kukuljević–Rajterov mlin.&lt;/ref&gt;

=== Southern branch ===
In the regional context, the family name appears in its localized form as ''Franjo Rajter'', reflecting linguistic adaptation within a multilingual imperial environment while preserving continuity of lineage identity.

Franz von Reiter (Franjo Rajter) was the father of seven children, consisting of six daughters and one son, '''Franjo Rajter''' (born 1917), who continued the family line.

The latter attended the [[wikipedia:University_of_Zagreb|University of Zagreb]], where he completed studies in law and economics, reflecting the continued emphasis on higher education and professional formation within the lineage. In his later career, he served as a managing director within the national railway system of Yugoslavia ([[wikipedia:Yugoslav_Railways|Jugoslavenske Željeznice]]), maintaining the tradition of institutional responsibility characteristic of the House.

== Coat of arms ==
[[File:Reiter_v_Ritterfels_archival.jpg|right|thumb|220x220px|Archival depiction of the coat of arms attributed to Franz Reiter von Ritterfels (1692–1702), Tiroler Landesmuseen Ferdinandeum.]]
[[File:Reiter_v_Ritterfels_reconstruction.png|right|thumb|226x226px|Modern artistic reconstruction based on archival reference, for illustrative purposes only.]]
A coat of arms associated with the name ''Reiter von Ritterfels'' is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

The armorial entry attributes the coat of arms to '''Franz Reiter von Ritterfels''' in his capacity as ''Pflegsverwalter'' at Lengberg. The surviving depiction reflects the heraldic style of Tyrol in the late 17th century, including a shield and helmet with crest.

No formal blazon is recorded in the archival entry. The depiction suggests a quartered shield featuring a mounted rider and a lion, surmounted by a helmet with crest, consistent with heraldic conventions of the period.

== Tradition and identity ==
The House of Reiter is characterized by a tradition rooted in:

* military discipline
* structured education
* administrative and institutional service
* restraint in outward expression of status

Rather than defining itself through hereditary rank alone, the lineage expresses identity through continuity of conduct, responsibility, and adherence to a disciplined cultural framework.

This reflects the broader model of the imperial service class, in which authority and standing were closely linked to function, education, and professional conduct.

== Motto ==
''Disciplina et Virtus''  (&quot;Discipline and Virtue&quot;)

== References ==
&lt;references /&gt;</text>
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{| class=&quot;wikitable&quot; style=&quot;float:right; width:300px; margin-left:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:center; font-size:120%;&quot; |Reiter Family
|-
| colspan=&quot;2&quot; style=&quot;text-align:center;&quot; |[[File:Reiter_v_Ritterfels_reconstruction.png|205x205px]]
|-
!Origin
|Tyrol, [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]]
|-
!Type
|Service lineage
|-
!Estate
|Lengberg (historical association)
|-
!Traditions
|Military, administrative, educational service
|-
!Motto
|''Disciplina et Virtus''
(&quot;Discipline and Virtue&quot;)
|}
The '''Reiter''' family is associated with traditions of imperial service within the [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]], particularly in the fields of military, administrative, and educational activity. The name is historically connected with Tyrol, where individuals bearing the surname ''Reiter'' are recorded in positions of judicial, military, and territorial authority from the 16th century onward.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entries for Reiter family (Lienz, Klausen, Innsbruck).&lt;/ref&gt;

The lineage developed within a tradition defined by disciplined service, military formation, and structured education, later associated with Vienna as the imperial center and subsequently extended into the southern regions of the Monarchy.

== History ==

=== Tyrolean origins ===
The earliest origins of the Reiter family are associated with Tyrol, a region of considerable strategic, military, and administrative importance within the Habsburg lands, serving as a vital corridor between the Alpine territories and northern Italy.

From the late 16th century onward, individuals bearing the name ''Reiter'' appear in positions of authority within the structures of local governance and territorial administration. Documented roles include ''Stadtrichter'' (city judges) in Lienz and Klausen, as well as a ''Stadthauptmann'' (city military commander) in Innsbruck.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'': Ruep Reiter (Stadt-Richter zu Lienz, 1580–1582, 1591–1593); Gabriel Reiter (Stadtrichter in Klausen, 1625); Michael Reiter (former Stadthauptmann in Innsbruck, c. 1650).&lt;/ref&gt;

These offices were integral to the functioning of the early modern state. The ''Stadtrichter'' exercised judicial authority and local administrative oversight, while the ''Stadthauptmann'' was responsible for urban defense, military organization, and coordination with regional command structures.

The recurrence of such appointments across different Tyrolean centers indicates the presence of the name within a service-oriented administrative and military stratum, in which advancement was based on discipline, reliability, and demonstrated competence within institutional frameworks.

This environment formed the early foundation of the lineage’s association with structured authority, military readiness, and civic responsibility.

=== Reiter von Ritterfels ===
A defining historical expression of the lineage is associated with '''Franz Reiter von Ritterfels''', recorded between 1692 and 1702 as ''Pflegsverwalter'' at Lengberg.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entry for Franz Reiter v. Ritterfels.&lt;/ref&gt;

This office represented a territorial administrative function combining executive governance, judicial authority, fiscal oversight, and supervision of regional order within a defined jurisdiction.

The ''Pflegsverwalter'' operated as a direct representative of territorial authority, responsible for maintaining legal order, administering estates, and overseeing local administration on behalf of higher governing structures.

The designation ''von Ritterfels'' reflects an association with territorial identity and administrative responsibility within the imperial system.

A coat of arms attributed to Franz Reiter von Ritterfels is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

In later family usage, the shortened form “von Reiter” emerged as a preserved expression of this heritage, retaining its association with service, discipline, and administrative function.

=== Viennese phase and continuation of imperial service tradition ===
During the mid to late 19th century, a branch of the Reiter family is associated with Vienna, reflecting its integration into the military and administrative structures of the Habsburg Monarchy.

A documented representative of this branch is '''Franz von Reiter''', an officer of the imperial army whose service is recorded in the ''Militär-Schematismus'' between 1850 and 1876.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850–1876.&lt;/ref&gt;

He entered service as an ''Unterleutnant'' in the 38th Infantry Regiment and advanced through the ranks to ''Oberleutnant'', later attaining the rank of ''Hauptmann 1. Classe'' in the 32nd Hungarian Infantry Regiment.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850, 1855, 1859, 1868; ''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1869–1876.&lt;/ref&gt;

His assignments included service in Lombardy, Lower Austria, Hungary, and the Adriatic regions, reflecting the geographical mobility characteristic of imperial officers.&lt;ref&gt;''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1871–1876.&lt;/ref&gt;

His son, '''Franz von Reiter''' (born 1878 in Vienna), served in the imperial army and subsequently relocated to the southern territories of the Monarchy, where he became established in the Ivanec region. By 1906, he was the proprietor of a mill known locally as ''Rajterov mlin''.&lt;ref&gt;Ivanec regional historical materials, Kukuljević–Rajterov mlin.&lt;/ref&gt;

=== Southern branch ===
In the regional context, the family name appears in its localized form as ''Franjo Rajter'', reflecting linguistic adaptation within a multilingual imperial environment while preserving continuity of lineage identity.

Franz von Reiter (Franjo Rajter) was the father of seven children, consisting of six daughters and one son, '''Franjo Rajter''' (born 1917), who continued the family line.

The latter attended the [[wikipedia:University_of_Zagreb|University of Zagreb]], where he completed studies in law and economics, reflecting the continued emphasis on higher education and professional formation within the lineage. In his later career, he served as a managing director within the national railway system of Yugoslavia ([[wikipedia:Yugoslav_Railways|Jugoslavenske Željeznice]]), maintaining the tradition of institutional responsibility characteristic of the House.

== Coat of arms ==
[[File:Reiter_v_Ritterfels_archival.jpg|right|thumb|220x220px|Archival depiction of the coat of arms attributed to Franz Reiter von Ritterfels (1692–1702), Tiroler Landesmuseen Ferdinandeum.]]
[[File:Reiter_v_Ritterfels_reconstruction.png|right|thumb|226x226px|Modern artistic reconstruction based on archival reference, for illustrative purposes only.]]
A coat of arms associated with the name ''Reiter von Ritterfels'' is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

The armorial entry attributes the coat of arms to '''Franz Reiter von Ritterfels''' in his capacity as ''Pflegsverwalter'' at Lengberg. The surviving depiction reflects the heraldic style of Tyrol in the late 17th century, including a shield and helmet with crest.

No formal blazon is recorded in the archival entry. The depiction suggests a quartered shield featuring a mounted rider and a lion, surmounted by a helmet with crest, consistent with heraldic conventions of the period.

== Tradition and identity ==
The House of Reiter is characterized by a tradition rooted in:

* military discipline
* structured education
* administrative and institutional service
* restraint in outward expression of status

Rather than defining itself through hereditary rank alone, the lineage expresses identity through continuity of conduct, responsibility, and adherence to a disciplined cultural framework.

This reflects the broader model of the imperial service class, in which authority and standing were closely linked to function, education, and professional conduct.

== Motto ==
''Disciplina et Virtus''  (&quot;Discipline and Virtue&quot;)

== References ==
&lt;references /&gt;

[[Category:History]]</text>
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{| class=&quot;wikitable&quot; style=&quot;float:right; width:300px; margin-left:15px;&quot;
! colspan=&quot;2&quot; style=&quot;text-align:center; font-size:120%;&quot; |Reiter Family
|-
| colspan=&quot;2&quot; style=&quot;text-align:center;&quot; |[[File:Reiter_v_Ritterfels_reconstruction.png|205x205px]]
|-
!Origin
|Tyrol, [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]]
|-
!Type
|Service lineage
|-
!Estate
|Lengberg (historical association)
|-
!Traditions
|Military, administrative, educational service
|-
!Motto
|''Disciplina et Virtus''
(&quot;Discipline and Virtue&quot;)
|}
The '''Reiter''' family is associated with traditions of imperial service within the [[wikipedia:Habsburg_monarchy|Habsburg Monarchy]], particularly in the fields of military, administrative, and educational activity. The name is historically connected with Tyrol, where individuals bearing the surname ''Reiter'' are recorded in positions of judicial, military, and territorial authority from the 16th century onward.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entries for Reiter family (Lienz, Klausen, Innsbruck).&lt;/ref&gt;

The lineage developed within a tradition defined by disciplined service, military formation, and structured education, later associated with Vienna as the imperial center and subsequently extended into the southern regions of the Monarchy.

== History ==

=== Tyrolean origins ===
The earliest origins of the Reiter family are associated with Tyrol, a region of considerable strategic, military, and administrative importance within the Habsburg lands, serving as a vital corridor between the Alpine territories and northern Italy.

From the late 16th century onward, individuals bearing the name ''Reiter'' appear in positions of authority within the structures of local governance and territorial administration. Documented roles include ''Stadtrichter'' (city judges) in Lienz and Klausen, as well as a ''Stadthauptmann'' (city military commander) in Innsbruck.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'': Ruep Reiter (Stadt-Richter zu Lienz, 1580–1582, 1591–1593); Gabriel Reiter (Stadtrichter in Klausen, 1625); Michael Reiter (former Stadthauptmann in Innsbruck, c. 1650).&lt;/ref&gt;

These offices were integral to the functioning of the early modern state. The ''Stadtrichter'' exercised judicial authority and local administrative oversight, while the ''Stadthauptmann'' was responsible for urban defense, military organization, and coordination with regional command structures.

The recurrence of such appointments across different Tyrolean centers indicates the presence of the name within a service-oriented administrative and military stratum, in which advancement was based on discipline, reliability, and demonstrated competence within institutional frameworks.

This environment formed the early foundation of the lineage’s association with structured authority, military readiness, and civic responsibility.

=== Reiter von Ritterfels ===
A defining historical expression of the lineage is associated with '''Franz Reiter von Ritterfels''', recorded between 1692 and 1702 as ''Pflegsverwalter'' at Lengberg.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entry for Franz Reiter v. Ritterfels.&lt;/ref&gt;

This office represented a territorial administrative function combining executive governance, judicial authority, fiscal oversight, and supervision of regional order within a defined jurisdiction.

The ''Pflegsverwalter'' operated as a direct representative of territorial authority, responsible for maintaining legal order, administering estates, and overseeing local administration on behalf of higher governing structures.

The designation ''von Ritterfels'' reflects an association with territorial identity and administrative responsibility within the imperial system.

A coat of arms attributed to Franz Reiter von Ritterfels is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

In later family usage, the shortened form “von Reiter” emerged as a preserved expression of this heritage, retaining its association with service, discipline, and administrative function.

=== Viennese phase and continuation of imperial service tradition ===
During the mid to late 19th century, a branch of the Reiter family is associated with Vienna, reflecting its integration into the military and administrative structures of the Habsburg Monarchy.

A documented representative of this branch is '''Franz von Reiter''', an officer of the imperial army whose service is recorded in the ''Militär-Schematismus'' between 1850 and 1876.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850–1876.&lt;/ref&gt;

He entered service as an ''Unterleutnant'' in the 38th Infantry Regiment and advanced through the ranks to ''Oberleutnant'', later attaining the rank of ''Hauptmann 1. Classe'' in the 32nd Hungarian Infantry Regiment.&lt;ref&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna, 1850, 1855, 1859, 1868; ''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1869–1876.&lt;/ref&gt;

His assignments included service in Lombardy, Lower Austria, Hungary, and the Adriatic regions, reflecting the geographical mobility characteristic of imperial officers.&lt;ref&gt;''Kaiserlich-königlicher Militär-Schematismus'', Vienna, 1871–1876.&lt;/ref&gt;

His son, '''Franz von Reiter''' (born 1878 in Vienna), served in the imperial army and subsequently relocated to the southern territories of the Monarchy, where he became established in the Ivanec region. By 1906, he was the proprietor of a mill known locally as ''Rajterov mlin''.&lt;ref&gt;Ivanec regional historical materials, Kukuljević–Rajterov mlin.&lt;/ref&gt;

=== Southern branch ===
In the regional context, the family name appears in its localized form as ''Franjo Rajter'', reflecting linguistic adaptation within a multilingual imperial environment while preserving continuity of lineage identity.

Franz von Reiter (Franjo Rajter) was the father of seven children, consisting of six daughters and one son, '''Franjo Rajter''' (born 1917), who continued the family line.

The latter attended the [[wikipedia:University_of_Zagreb|University of Zagreb]], where he completed studies in law and economics, reflecting the continued emphasis on higher education and professional formation within the lineage. In his later career, he served as a managing director within the national railway system of Yugoslavia ([[wikipedia:Yugoslav_Railways|Jugoslavenske Željeznice]]), maintaining the tradition of institutional responsibility characteristic of the House.

== Coat of arms ==
[[File:Reiter_v_Ritterfels_archival.jpg|right|thumb|220x220px|Archival depiction of the coat of arms attributed to Franz Reiter von Ritterfels (1692–1702), Tiroler Landesmuseen Ferdinandeum.]]
[[File:Reiter_v_Ritterfels_reconstruction.png|right|thumb|226x226px|Modern artistic reconstruction based on archival reference, for illustrative purposes only.]]
A coat of arms associated with the name ''Reiter von Ritterfels'' is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76: ''Reiter v. Ritterfels Franz'', Pflegsverwalter zu Lengberg (1692–1702).&lt;/ref&gt;

The armorial entry attributes the coat of arms to '''Franz Reiter von Ritterfels''' in his capacity as ''Pflegsverwalter'' at Lengberg. The surviving depiction reflects the heraldic style of Tyrol in the late 17th century, including a shield and helmet with crest.

No formal blazon is recorded in the archival entry. The depiction suggests a quartered shield featuring a mounted rider and a lion, surmounted by a helmet with crest, consistent with heraldic conventions of the period.

== Tradition and identity ==
The House of Reiter is characterized by a tradition rooted in:

* military discipline
* structured education
* administrative and institutional service
* restraint in outward expression of status

Rather than defining itself through hereditary rank alone, the lineage expresses identity through continuity of conduct, responsibility, and adherence to a disciplined cultural framework.

This reflects the broader model of the imperial service class, in which authority and standing were closely linked to function, education, and professional conduct.

== Motto ==
''Disciplina et Virtus''  (&quot;Discipline and Virtue&quot;)

== References ==
&lt;references /&gt;

[[Category:History]]</text>
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The '''Reiter''' family is associated with traditions of service within the Habsburg Monarchy, particularly in the fields of military, administrative, and educational activity. The name is historically connected with Tyrol, where individuals bearing the surname ''Reiter'' are recorded in positions of judicial, military, and territorial responsibility from the 16th century onward.&lt;ref name=&quot;Ferdinandeum_general&quot;&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entries for Reiter family (Lienz, Klausen, Innsbruck, Ynsprugg/Innsbruck, Lengberg).&lt;/ref&gt;

The available material suggests the development of a pattern of service characterized by disciplined administrative function, military organization, and structured education within the institutional framework of the Habsburg Monarchy, a composite state marked by decentralized governance and service-based advancement.&lt;ref name=&quot;Ingrao&quot;&gt;Charles W. Ingrao, ''The Habsburg Monarchy, 1618–1815'', Cambridge University Press.&lt;/ref&gt;&lt;ref name=&quot;Judson&quot;&gt;Pieter M. Judson, ''The Habsburg Empire: A New History'', Harvard University Press, 2016.&lt;/ref&gt; These associations later extend toward Vienna as the imperial center and into the southern regions of the Monarchy.

Presented as a research-based learning resource, the page explores patterns of administrative and military service in the Habsburg Monarchy using the Reiter surname as a case study. Rather than reconstructing a continuous genealogy, the focus lies on identifying recurring functional roles and institutional integration across different historical contexts.

In addition to published and archival materials, the reconstruction is informed by private sources, including family-held materials such as parish records, educational documentation (e.g., enrollment records, diplomas, and institutional certificates), and historical correspondence. These provide supplementary context but are not fully represented in publicly accessible archives and are therefore treated with methodological caution.

The account does not attempt to represent the full historical extent of the surname ''Reiter'', which occurs in multiple independent contexts across Central Europe. Such an approach allows for the examination of broader structural patterns through limited but traceable historical evidence.

== History ==

=== Tyrolean origins ===
The earliest occurrences of the name are associated with Tyrol, a region of considerable strategic, military, and administrative importance within the Habsburg lands. Positioned along key Alpine transit routes linking Central Europe with northern Italy, Tyrol functioned both as a defensive frontier and as a corridor of economic, political, and administrative exchange.&lt;ref name=&quot;Ingrao&quot; /&gt;

From the late 16th century onward, individuals bearing the name ''Reiter'' appear in documented positions of authority within local governance and territorial administration. Recorded roles include ''Stadtrichter'' (city judge, combining judicial authority with administrative oversight) in Lienz and Klausen, as well as a ''Stadthauptmann'' (urban military commander responsible for defense and coordination) in Innsbruck.&lt;ref name=&quot;Ferdinandeum_roles&quot;&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'': Ruep Reiter (Stadt-Richter zu Lienz, 1580–1582, 1591–1593); Gabriel Reiter (Stadtrichter in Klausen, 1625); Michael Reiter (former Stadthauptmann in Ynsprugg/Innsbruck, c. 1648–1650).&lt;/ref&gt;

These offices formed part of a governance structure in which judicial, fiscal, and military responsibilities were not rigidly separated but instead overlapped within the framework of territorial lordship. Local officials operated within a layered administrative system linking municipalities, regional estates (''Landstände''), and central Habsburg authority. In this capacity, such figures acted as intermediaries, translating territorial and dynastic directives into local practice while maintaining regional stability.&lt;ref name=&quot;Ingrao&quot; /&gt;

The recurrence of these roles across multiple Tyrolean centers suggests a sustained integration of the name within a service-oriented administrative milieu, where advancement depended on demonstrated competence, reliability, and the ability to operate within established institutional frameworks. This reflects broader processes of early modern state formation, in which governance relied on locally embedded yet centrally aligned officeholders.&lt;ref name=&quot;Ingrao&quot; /&gt;

=== Reiter von Ritterfels ===

[[File:Reiter_v_Ritterfels_archival.jpg|right|thumb|230px|Archival depiction attributed to Franz Reiter von Ritterfels (1692–1702), Tiroler Landesmuseen Ferdinandeum.]]

A significant archival entry records '''Franz Reiter von Ritterfels''' as ''Pflegsverwalter'' (territorial administrator) at Lengberg between 1692 and 1702.&lt;ref name=&quot;Ferdinandeum_Ritterfels&quot;&gt;Tiroler Landesmuseen Ferdinandeum, ''Fischnaler Wappenkartei'', entry for Franz Reiter v. Ritterfels; Lengberg (1692; 1702).&lt;/ref&gt;

The office of ''Pflegsverwalter'' combined executive authority, judicial competence, fiscal management, and oversight of territorial order within a defined jurisdiction. Such officials functioned as direct representatives of territorial authority, responsible for enforcing legal norms, managing estates, and maintaining administrative continuity.&lt;ref name=&quot;Ingrao&quot; /&gt;

The designation ''von Ritterfels'' appears in connection with this entry and may reflect a territorial, administrative, or locational association within the imperial system rather than a fixed hereditary title. In later usage, shortened forms such as ''von Reiter'' appear and may indicate the adaptation or reinterpretation of such identifiers within evolving social and bureaucratic contexts.

=== Use of the designation &quot;von Reiter&quot; ===
The use of the particle ''von'' in Central European naming conventions has historically been fluid. While often associated with forms of social distinction, it has also been employed to denote geographic origin, administrative affiliation, or adopted identifiers within military and bureaucratic environments.

In this context, the designation ''von Reiter'' is best understood as reflecting patterns of documented usage rather than constituting definitive evidence of formally recognized status, illustrating the flexibility of naming practices within the Habsburg administrative world.

=== Heraldic material ===

[[File:Reiter_v_Ritterfels_reconstruction.png|right|thumb|230px|Modern reconstruction based on archival reference, for illustrative purposes.]]

A coat of arms associated with the name ''Reiter von Ritterfels'' is preserved in the collections of the Tiroler Landesmuseen Ferdinandeum.&lt;ref name=&quot;Ferdinandeum_heraldry&quot;&gt;Tiroler Landesmuseen Ferdinandeum, Wappensammlung (Josef Oberforcher), Blatt 76.&lt;/ref&gt;

The archival entry attributes the coat of arms to Franz Reiter von Ritterfels in his capacity as ''Pflegsverwalter'' at Lengberg. The depiction reflects the heraldic conventions of late 17th-century Tyrol, including shield, helmet, and crest elements.

No formal blazon is recorded in the available material. The reconstructed depiction suggests a quartered shield featuring a mounted rider and a lion, consistent with symbolic motifs associated with mobility, authority, and strength.

No historically attested motto is recorded. In later interpretive usage, the phrase ''Disciplina et Virtus'' (&quot;Discipline and Virtue&quot;) has been associated with the reconstructed context as a conceptual summary derived from patterns of administrative and military service. References to similar formulations appear in private correspondence; however, these remain fragmentary and do not constitute formal heraldic attribution.

=== Viennese phase and continuation of service context ===
During the mid to late 19th century, individuals bearing the name are associated with Vienna, reflecting participation in the military and administrative structures of the Habsburg Monarchy, whose capital functioned as the central node of imperial governance, bureaucracy, and military command.&lt;ref name=&quot;Beller&quot;&gt;Steven Beller, ''The Habsburg Monarchy, 1815–1918'', Cambridge University Press, 2018.&lt;/ref&gt;&lt;ref name=&quot;Judson&quot; /&gt;

A documented representative is '''Franz von Reiter''', recorded in the ''Militär-Schematismus'' (official military register) between 1850 and 1876.&lt;ref name=&quot;Schematismus1&quot;&gt;''Militär-Schematismus des österreichischen Kaiserthumes'', Vienna: k.k. Hof- und Staatsdruckerei, various years, 1850–1876.&lt;/ref&gt;&lt;ref name=&quot;Schematismus2&quot;&gt;''Kaiserlich-königlicher Militär-Schematismus'', Vienna, various years.&lt;/ref&gt;

He entered service as an ''Unterleutnant'' (second lieutenant) in the 38th Infantry Regiment and advanced to ''Oberleutnant'' (first lieutenant), later attaining the rank of ''Hauptmann 1. Classe'' (captain, first class) in the 32nd Hungarian Infantry Regiment.

The infantry regiments (''Infanterieregimenter'') of the Habsburg army were organized as numbered units tied to specific recruitment districts (''Ergänzungsbezirke''), though their operational deployment frequently extended far beyond these regions. The surviving military archival framework also reflects this territorial logic, with personnel and recruiting records often organized by recruiting district, regiment, or crown land.&lt;ref&gt;Kriegsarchiv, Austrian State Archives, “Family research – Kriegsarchiv”.&lt;/ref&gt;

These regiments were composed of soldiers drawn from multiple ethnic, linguistic, and regional backgrounds, illustrating the multinational character of the imperial army. Officers were regularly transferred between units and stationed across different crown lands, contributing to a shared institutional culture based on standardized training, discipline, and loyalty to central command rather than local affiliation.&lt;ref name=&quot;Beller&quot; /&gt;&lt;ref name=&quot;Judson&quot; /&gt;

Units of this type were regularly engaged in major military theaters of the 19th century, including campaigns associated with the Italian Wars of Independence and the suppression of revolutionary movements in 1848–1849. Although direct participation of specific individuals in particular engagements cannot always be verified, service within such regiments situates them within the broader framework of imperial military operations.&lt;ref name=&quot;Beller&quot; /&gt;

This period coincided with profound structural transformations, including the Revolutions of 1848 and the Austro-Hungarian Compromise of 1867, which reshaped political authority, administrative organization, and military command structures. The army underwent processes of modernization and professionalization, introducing more uniform training systems, clearer hierarchies, and increasingly centralized coordination across diverse territories.&lt;ref name=&quot;Beller&quot; /&gt;&lt;ref name=&quot;Judson&quot; /&gt;

Within this evolving system, military service provided a pathway for advancement that relied less on hereditary privilege and more on education, discipline, and demonstrated competence. The mobility of officers across different regions contributed to the development of a cohesive imperial administrative and military culture despite the growing complexity of the dual monarchy.&lt;ref name=&quot;Beller&quot; /&gt;

A later individual, '''Franz von Reiter''' (born 1878 in Vienna), is associated with relocation to the southern territories of the Monarchy, where he is recorded in regional historical materials as having acquired a mill in the Ivanec area by 1906.&lt;ref name=&quot;Ivanec&quot;&gt;Regional historical materials for Ivanec, including references to ''Rajterov mlin'' in local historiography (Kukuljević collection and related studies, Varaždin region).&lt;/ref&gt; While a connection to military service may be suggested by contextual patterns, no specific ranks or functions are documented in the available sources.

=== Southern branch ===

==== Linguistic and regional adaptation ====
In the regional context, the surname appears in the localized form ''Rajter'', reflecting phonetic adaptation within South Slavic linguistic environments.

The transformation from ''Reiter'' to ''Rajter'' follows identifiable phonological patterns characteristic of the adaptation of German-language surnames into South Slavic linguistic systems. The German diphthong ''ei'' is regularly rendered as ''aj'', while the consonantal structure remains largely preserved, resulting in a form that is both phonetically consistent with the original and integrated into local pronunciation norms.

Such transformations were not isolated phenomena but part of a broader pattern within multilingual regions of the Habsburg Monarchy, where administrative, military, and civilian populations interacted across linguistic boundaries. Surnames were frequently adapted in parish records, civil registries, and everyday usage, reflecting both practical pronunciation and evolving linguistic standardization.

This process intensified during the late 19th and early 20th centuries, particularly in regions such as northern Croatia, where administrative bilingualism gradually gave way to the increased standardization of South Slavic languages in official and educational contexts. Following the dissolution of Austria-Hungary in 1918, newly formed national frameworks introduced more consistent orthographic and linguistic norms, further reinforcing localized surname forms.&lt;ref name=&quot;Beller&quot; /&gt;&lt;ref name=&quot;Judson&quot; /&gt;

==== Rajterov mlin and regional integration ====
The reference to ''Rajterov mlin'' (also recorded as ''Kukuljević–Rajterov mlin'' and in some instances as ''Reiterov mlin'' in regional historical materials) provides a concrete example of this transition, illustrating how a name associated with imperial administrative and military contexts became embedded within a regional economic setting.&lt;ref name=&quot;IvanecMlinovi&quot;&gt;Ivanečki mlinovi (Ivanec watermills), regional historical overview.&lt;/ref&gt;

In this context, '''Franz von Reiter (born 1878 in Vienna)''' is associated with the Ivanec area in the early 20th century. Within the regional South Slavic linguistic environment, his name appears in the localized form '''Franjo Rajter''', reflecting both phonetic adaptation and integration into local cultural and administrative usage.

This transformation from ''Franz von Reiter'' to ''Franjo Rajter'' corresponds to broader patterns of linguistic and social adaptation characteristic of the late Habsburg and immediate post-imperial period, in which individuals moving between German-speaking and South Slavic regions frequently adopted localized forms of personal and family names in accordance with prevailing administrative, educational, and social norms.

Regional documentation identifies the mill as one of several significant milling installations, explicitly referring to a ''Kukuljević–Rajterov mlin'', indicating either successive ownership, shared association, or historiographical layering of naming conventions within local records.&lt;ref name=&quot;IvanecMlinovi&quot; /&gt;

The variation between ''Reiterov'', ''Rajterov'', and ''Kukuljević–Rajterov'' reflects the multilingual administrative and historiographical environment of the late 19th and early 20th centuries. German-derived naming conventions, South Slavic phonetic adaptation, and later standardized Croatian historiography intersect in these forms, preserving multiple layers of linguistic and social identity within the documentary record.

In this context, the mill functioned not only as a private enterprise but also as a structurally important component of the regional agrarian economy. Watermills in this period served as key infrastructural nodes, linking agricultural production (grain cultivation), processing (milling), and local distribution networks. Their operation was often tied to landownership, tenancy relations, and local administrative frameworks, embedding them within both economic and social hierarchies.

The association of Franz von Reiter (Franjo Rajter) with this environment reflects the transformation of identity from an imperial service-oriented context into a localized economic and social presence. It also corresponds to a broader pattern observed in post-imperial regions, where individuals and families formerly connected to administrative or military structures adapted to new roles within regional economic systems following the restructuring of political authority after the dissolution of Austria-Hungary.

==== Family continuity and 20th-century development ====
Franz von Reiter (Franjo Rajter) is recorded in family-related materials as having had seven children—six daughters and one son. His son, '''Franjo Rajter (born 1917)''', represents the continuation of the localized form of the surname within the regional context.

He is associated in family-held and related materials with studies in law and economics at the University of Zagreb and with subsequent professional activity within Jugoslav Railways (''Jugoslavenske Željeznice''), reflecting continued participation in structured institutional environments requiring formal education and administrative competence.

Within the context of the Yugoslav state, such positions were embedded within centralized administrative and infrastructural systems, continuing earlier patterns of institutional integration, albeit within a transformed political and organizational framework.

==== Migration and post-imperial transformation ====
Migration patterns further reflect adaptation to changing historical conditions. During the 20th century, individuals associated with this branch are documented in family-related and regional materials as emigrating to countries including the United States, Australia, and Argentina. These movements correspond to broader Central and Eastern European migration trends driven by economic factors, political change, and the reconfiguration of state structures.

In some cases, later return movements toward Austria, particularly Vienna, are also noted, indicating continued connections with earlier administrative and cultural centers of the Habsburg world.

These developments may be situated within the broader context of post-imperial transformation, including the dissolution of imperial administrative systems, the emergence of nation-states, and the redefinition of identity markers such as language, surname, and institutional affiliation. In this environment, the adaptation from ''Reiter'' to ''Rajter'' represents both a linguistic adjustment and a reflection of integration into new socio-political frameworks.
== Methodology ==

This learning resource is based on a combination of primary, secondary, and supplementary private materials, analyzed using a case study approach focused on the occurrence of the surname ''Reiter'' within the administrative and military structures of the Habsburg Monarchy.

Primary sources include archival records and official registers, while secondary sources provide regional and historical context. Supplementary private materials, including parish documentation, educational records, and correspondence, are used cautiously to provide continuity where public records are incomplete.

The methodological framework corresponds to a prosopographical and microhistorical approach, examining individual instances as part of broader patterns of institutional service, social mobility, and administrative integration.&lt;ref name=&quot;Judson&quot; /&gt;

Interpretation distinguishes between documented evidence, contextual inference, and later interpretive elements, acknowledging the limitations inherent in fragmentary and heterogeneous source material.

== Interpretive analysis ==

The available evidence suggests a pattern of functional continuity characterized by repeated integration into military, administrative, and educational institutions across different historical periods. These patterns reflect broader dynamics of governance, professionalization, and mobility within the Habsburg Monarchy rather than a fixed hereditary identity.&lt;ref name=&quot;Ingrao&quot; /&gt;&lt;ref name=&quot;Beller&quot; /&gt;&lt;ref name=&quot;Judson&quot; /&gt;

== References ==
&lt;references /&gt;

[[Category:History]]</text>
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      <comment>Created page with &quot;TBD  studying pre-calc &amp; calc for fun, anything useful would be put here and hopefully integrated in the mainspace.&quot;</comment>
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      <text bytes="115" sha1="5yr0jz1smcoare3o1z0yqawrn451o4z" xml:space="preserve">TBD

studying pre-calc &amp; calc for fun, anything useful would be put here and hopefully integrated in the mainspace.</text>
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      <comment>cat(s)</comment>
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      <text bytes="165" sha1="p79aomi27meecyplb5zuzfcimmqkoe5" xml:space="preserve">TBD

studying pre-calc &amp; calc for fun, anything useful would be put here and hopefully integrated in the mainspace.

[[Category:Atcovi's Work]]
[[Category:Calculus]]</text>
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      <comment>Created page with &quot;* [https://scholar.google.com/citations?user=A5JbI2QAAAAJ&amp;hl=en&amp;oi=ao Google Scholar] * [https://orcid.org/0000-0003-0710-4550 ORCID] * [http://www.canberra.edu.au/about-uc/faculties/health/courses/psychology/tabs/staff-profiles/staff-profiles/academic-staff/neill-james Pure] (University of Canberra) &lt;!-- {{note|Material from [http://ucspace.canberra.edu.au/display/~s613374/Research research] (ucspace) to be transferred here.}} --&gt;&quot;</comment>
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      <text bytes="435" sha1="pcz8q0lkehlgvh19f8do5wos1a6wvet" xml:space="preserve">* [https://scholar.google.com/citations?user=A5JbI2QAAAAJ&amp;hl=en&amp;oi=ao Google Scholar]
* [https://orcid.org/0000-0003-0710-4550 ORCID]
* [http://www.canberra.edu.au/about-uc/faculties/health/courses/psychology/tabs/staff-profiles/staff-profiles/academic-staff/neill-james Pure] (University of Canberra)
&lt;!-- {{note|Material from [http://ucspace.canberra.edu.au/display/~s613374/Research research] (ucspace) to be transferred here.}} --&gt;</text>
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      <comment>Tidy</comment>
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* [https://scholar.google.com/citations?user=A5JbI2QAAAAJ&amp;hl=en&amp;oi=ao Google Scholar]
* [https://orcid.org/0000-0003-0710-4550 ORCID]
* [https://researchprofiles.canberra.edu.au/en/persons/james-neill PURE] (University of Canberra)</text>
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      <comment>Created page with &quot;= Transcript Archive Analysis =  == Sequential Overview == * Total of 7 conversations across the archive * Date range spans from January 28, 2026 to April 1, 2026 * First prompt was exploratory: “what happens when i think” * Most recent prompt involved structured academic synthesis * Usage shows bursts of activity followed by noticeable gaps  ----  == Prompting Taxonomy == * 86% of prompts focus on exploration, structuring, and synthesis * Equal distribution across c...&quot;</comment>
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      <text bytes="2824" sha1="194wjh35j8kze7w68nwqxabyi655gwl" xml:space="preserve">= Transcript Archive Analysis =

== Sequential Overview ==
* Total of 7 conversations across the archive
* Date range spans from January 28, 2026 to April 1, 2026
* First prompt was exploratory: “what happens when i think”
* Most recent prompt involved structured academic synthesis
* Usage shows bursts of activity followed by noticeable gaps

----

== Prompting Taxonomy ==
* 86% of prompts focus on exploration, structuring, and synthesis
* Equal distribution across concept exploration, structured output, and analysis (28.6% each)
* Minimal use of simple information lookup (14.3%)
* No usage of debugging, brainstorming, or feedback categories
* Prompts emphasize precision, formatting, and constraints

----

== Acceptance vs Pushback ==
* Rarely accepts first response as final output
* Pushes refinement through added constraints rather than contradiction
* Uses follow-ups to improve structure and accuracy
* Accepts core ideas but improves presentation and rigor
* Demonstrates controlled, iterative improvement rather than rejection

----

== Follow-Up Behavior ==
* Frequently asks follow-up prompts within the same topic
* Each follow-up increases complexity or specificity
* Uses follow-ups as a tool for iteration, not clarification
* Maintains continuity rather than switching topics
* Builds layered outputs over multiple steps

----

== Conversation Control ==
* Strongly directs conversation with explicit instructions
* Specifies format, structure, and output requirements
* Limits model creativity to meet exact expectations
* Rarely allows model to lead or reinterpret tasks
* Treats AI as an execution tool rather than a collaborator

----

== Exploration vs Goal Orientation ==
* Begins with brief conceptual exploration
* Quickly transitions into structured, goal-driven tasks
* Focuses on producing usable outputs rather than open discussion
* Minimizes time spent on abstract or open-ended thinking
* Prioritizes task completion over exploration

----

== Iteration Style ==
* Performs deep iteration within short timeframes
* Rapid refinement during active sessions
* Long gaps between separate work periods
* Each session builds toward a clear end product
* Shows structured workflow rather than continuous usage

----

== Behavioral Assessment ==
* Operates as a directive, high-control user
* Uses AI as a structured processing pipeline
* Focuses on output quality, formatting, and accuracy
* Avoids ambiguity by adding constraints
* Demonstrates consistent, goal-oriented interaction patterns

----

== Missing Behaviors ==
* Does not request critique or evaluation of outputs
* Rarely challenges correctness directly
* Does not engage in brainstorming or idea generation
* Minimal adversarial or “what’s wrong with this” prompts
* Limited use of open-ended or exploratory dialogue</text>
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    <title>DesignWriteStudio/Course/StudentPages/ChrisP/5.1 South America</title>
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      <comment>Created page with &quot;{{:DesignWriteStudio/SiteElements/Navbox}}  = South America: Sovereign States by Population and Area =  This page presents a data-based overview of the sovereign states in South America using Wikidata query results exported to TSV files. It includes a complete sortable table, top-five rankings by population and land area, a short comparative analysis of what those rankings suggest, and a brief discussion of data quality and limitations.&lt;ref&gt;{{cite web |title=Wikidata Que...&quot;</comment>
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= South America: Sovereign States by Population and Area =

This page presents a data-based overview of the sovereign states in South America using Wikidata query results exported to TSV files. It includes a complete sortable table, top-five rankings by population and land area, a short comparative analysis of what those rankings suggest, and a brief discussion of data quality and limitations.&lt;ref&gt;{{cite web |title=Wikidata Query Service query for South American sovereign states with population and area |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%20%20%20%23%20instance%20of%20sovereign%20state%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%20%20%20%20%20%20%20%20%23%20continent%20%3D%20South%20America%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation%3B%20%23%20population%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%20%20%20%20%20%20%20%23%20area%20in%20square%20kilometers%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service |access-date=2026-04-12}}&lt;/ref&gt;

== Complete table of sovereign states ==

The following table combines the population and area results for all South American sovereign states represented in the dataset.&lt;ref name=&quot;combined-query&quot; /&gt;

{| class=&quot;wikitable sortable&quot;
! Country
! Population

! Area (km²)
Argentina
-
Bolivia
-
Brazil
-
Chile
-
Colombia
-
Ecuador
-
Guyana
-
Paraguay
-
Peru
-
Suriname
-
Uruguay
-
Venezuela
}

== Top 5 by population ==

This ranking highlights the most populous sovereign states in South America based on the Wikidata population values in the dataset.&lt;ref&gt;{{cite web |title=Wikidata Query Service query for South American sovereign states by population |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%20%20%20%23%20instance%20of%20sovereign%20state%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%20%20%20%20%20%20%20%20%23%20continent%20%3D%20South%20America%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation.%20%23%20population%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service |access-date=2026-04-12}}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Population

! Area (km²)
1
-
2
-
3
-
4
-
5
}

== Top 5 by area ==

This ranking highlights the largest sovereign states in South America by area, measured in square kilometers.&lt;ref&gt;{{cite web |title=Wikidata Query Service query for South American sovereign states by area |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%20%20%20%23%20instance%20of%20sovereign%20state%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%20%20%20%20%20%20%20%20%23%20continent%20%3D%20South%20America%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%20%20%20%20%20%20%20%23%20area%20in%20square%20kilometers%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service |access-date=2026-04-12}}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Area (km²)

! Population
1
-
2
-
3
-
4
-
5
}

== Analysis: What the two top-5 lists argue ==

Taken together, the two top-five tables suggest that size and population are related in South America, but they are not the same thing. Brazil appears first on both lists, which argues that it is both territorially dominant and demographically dominant within the continent.&lt;ref name=&quot;combined-query&quot;&gt;{{cite web |title=Wikidata Query Service query for South American sovereign states with population and area |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%20%20%20%23%20instance%20of%20sovereign%20state%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%20%20%20%20%20%20%20%20%23%20continent%20%3D%20South%20America%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation%3B%20%23%20population%0A%20%20%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%20%20%20%20%20%20%20%23%20area%20in%20square%20kilometers%0A%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service |access-date=2026-04-12}}&lt;/ref&gt;

However, the lists also diverge in important ways. Colombia ranks second in population but fourth in area, which suggests a comparatively high concentration of people relative to land size among the continent’s largest countries. Bolivia, by contrast, appears in the top five by area but not in the top five by population, which argues that a large territorial footprint does not necessarily produce a similarly large population.&lt;ref name=&quot;combined-query&quot; /&gt;

Argentina and Peru appear on both lists, but in different positions. This suggests that they are important in both territorial and demographic terms, though not to the same degree. Venezuela appears in the population top five but not the area top five, reinforcing the idea that demographic significance can exist without being among the very largest states by land area.&lt;ref name=&quot;combined-query&quot; /&gt;

Overall, the two rankings argue that South America is shaped by one overwhelming outlier—Brazil—and then by a second tier of states whose demographic and territorial importance only partially overlap. In other words, population hierarchy and land-area hierarchy are connected, but they do not map perfectly onto one another.&lt;ref name=&quot;combined-query&quot; /&gt;

== Data quality and limitations ==

This dataset is useful for comparative overview, but it should be read with several limitations in mind.

The data comes from Wikidata, which is an open, collaboratively edited knowledge base. Values may change over time as items are updated.&lt;ref name=&quot;combined-query&quot; /&gt;
Population figures in Wikidata may represent estimates from different years, depending on the source attached to each item. That means the table supports comparison, but not necessarily strict same-year demographic analysis.
Area values are more stable than population values, but they can still differ depending on whether a source measures total area, land area, or updated boundary conventions.
The query filters for items that are both ''instance of sovereign state'' and located on the continent of South America. This is appropriate for a continent-level sovereign-state list, but it excludes non-sovereign territories and dependencies.
One area value in the dataset includes decimal precision (Ecuador at 257,204.27 km²), while most others appear as whole numbers. This reflects source formatting rather than a substantive analytical difference.

Because of these factors, the tables are best understood as a well-structured snapshot for broad comparison rather than a final authority on official statistical rankings.&lt;ref name=&quot;combined-query&quot; /&gt;

== References ==
&lt;references /&gt;

{{:DesignWriteStudio/SiteElements/Footer}}</text>
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= South America: Sovereign States by Population and Area =

This page presents sovereign states in South America using data exported from Wikidata query results. It includes a complete sortable table, two top-5 ranking tables, an analysis comparing the rankings, and a short data quality section.&lt;ref name=&quot;combined&quot;&gt;{{cite web
 |title=Wikidata Query Service: South American sovereign states with population and area
 |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service
 |access-date=2026-04-12
}}&lt;/ref&gt;

== Complete table of sovereign states ==

{| class=&quot;wikitable sortable&quot;
! Country
! Population
! Area (km²)
|-
| Argentina || 47,327,407 || 2,780,400
|-
| Bolivia || 12,244,159 || 1,098,581
|-
| Brazil || 213,421,037 || 8,515,767
|-
| Chile || 19,458,000 || 756,102
|-
| Colombia || 52,321,152 || 1,141,748
|-
| Ecuador || 16,938,986 || 257,204.27
|-
| Guyana || 878,674 || 214,970
|-
| Paraguay || 6,811,297 || 406,756
|-
| Peru || 33,726,000 || 1,285,216
|-
| Suriname || 563,402 || 163,270
|-
| Uruguay || 3,444,263 || 176,215
|-
| Venezuela || 31,250,306 || 912,050
|}

== Top 5 by population ==

The largest sovereign states in South America by population are the following.&lt;ref name=&quot;pop&quot;&gt;{{cite web
 |title=Wikidata Query Service: South American sovereign states with population
 |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service
 |access-date=2026-04-12
}}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Population
|-
| 1 || Brazil || 213,421,037
|-
| 2 || Colombia || 52,321,152
|-
| 3 || Argentina || 47,327,407
|-
| 4 || Peru || 33,726,000
|-
| 5 || Venezuela || 31,250,306
|}

== Top 5 by area ==

The largest sovereign states in South America by area are the following.&lt;ref name=&quot;area&quot;&gt;{{cite web
 |title=Wikidata Query Service: South American sovereign states with area
 |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service
 |access-date=2026-04-12
}}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Area (km²)
|-
| 1 || Brazil || 8,515,767
|-
| 2 || Argentina || 2,780,400
|-
| 3 || Peru || 1,285,216
|-
| 4 || Colombia || 1,141,748
|-
| 5 || Bolivia || 1,098,581
|}

== Analysis: What the two top-5 lists argue ==

The two rankings suggest that territorial size and population are related in South America, but they do not perfectly match. Brazil ranks first on both lists, which argues that it dominates the continent in both land area and population.&lt;ref name=&quot;combined&quot; /&gt;

The overlap between the two lists is substantial but incomplete. Brazil, Argentina, Peru, and Colombia appear on both lists, suggesting that these countries have both territorial and demographic weight within the continent.&lt;ref name=&quot;combined&quot; /&gt;

The differences are just as important. Venezuela appears in the top 5 by population but not in the top 5 by area, which suggests that a country can be demographically significant without being among the largest by land area. Bolivia appears in the top 5 by area but not in the top 5 by population, which suggests that large territory does not necessarily mean a proportionally large population.&lt;ref name=&quot;combined&quot; /&gt;

Together, the two lists argue that South America has one clear continental giant—Brazil—followed by a second tier of countries whose importance depends on whether the reader is measuring people or territory. The rankings therefore show that geographic scale and human concentration are connected, but they are not identical forms of significance.&lt;ref name=&quot;combined&quot; /&gt;

== Data quality ==

Several data-quality issues should be kept in mind when reading these tables.

* The values come from Wikidata, an open and collaboratively edited database, so figures can change over time as entries are updated.&lt;ref name=&quot;combined&quot; /&gt;
* Population figures may reflect estimates from different years, even when they are queried together in one result set.
* Area values are generally more stable than population values, but formatting can vary. For example, Ecuador appears with a decimal value of 257,204.27 km² while most other countries appear as whole numbers.
* The query is limited to items classified as sovereign states on the continent of South America, so it excludes non-sovereign territories and dependencies.
* The dataset is best understood as a comparative snapshot rather than a fixed official statistical publication.&lt;ref name=&quot;combined&quot; /&gt;

== References ==
&lt;references /&gt;

{{:DesignWriteStudio/SiteElements/Footer}}</text>
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        <id>2888076</id>
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      <minor />
      <comment>added [[Category:South America]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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= South America: Sovereign States by Population and Area =

This page presents sovereign states in South America using data exported from Wikidata query results. It includes a complete sortable table, two top-5 ranking tables, an analysis comparing the rankings, and a short data quality section.&lt;ref name=&quot;combined&quot;&gt;{{cite web
 |title=Wikidata Query Service: South American sovereign states with population and area
 |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service
 |access-date=2026-04-12
}}&lt;/ref&gt;

== Complete table of sovereign states ==

{| class=&quot;wikitable sortable&quot;
! Country
! Population
! Area (km²)
|-
| Argentina || 47,327,407 || 2,780,400
|-
| Bolivia || 12,244,159 || 1,098,581
|-
| Brazil || 213,421,037 || 8,515,767
|-
| Chile || 19,458,000 || 756,102
|-
| Colombia || 52,321,152 || 1,141,748
|-
| Ecuador || 16,938,986 || 257,204.27
|-
| Guyana || 878,674 || 214,970
|-
| Paraguay || 6,811,297 || 406,756
|-
| Peru || 33,726,000 || 1,285,216
|-
| Suriname || 563,402 || 163,270
|-
| Uruguay || 3,444,263 || 176,215
|-
| Venezuela || 31,250,306 || 912,050
|}

== Top 5 by population ==

The largest sovereign states in South America by population are the following.&lt;ref name=&quot;pop&quot;&gt;{{cite web
 |title=Wikidata Query Service: South American sovereign states with population
 |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Fpopulation%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP1082%20%3Fpopulation.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service
 |access-date=2026-04-12
}}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Population
|-
| 1 || Brazil || 213,421,037
|-
| 2 || Colombia || 52,321,152
|-
| 3 || Argentina || 47,327,407
|-
| 4 || Peru || 33,726,000
|-
| 5 || Venezuela || 31,250,306
|}

== Top 5 by area ==

The largest sovereign states in South America by area are the following.&lt;ref name=&quot;area&quot;&gt;{{cite web
 |title=Wikidata Query Service: South American sovereign states with area
 |url=https://query.wikidata.org/#SELECT%20%3Fcountry%20%3FcountryLabel%20%3Farea%20WHERE%20%7B%0A%20%20%3Fcountry%20wdt%3AP31%20wd%3AQ3624078%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP30%20wd%3AQ18%3B%0A%20%20%20%20%20%20%20%20%20wdt%3AP2046%20%3Farea.%0A%20%20SERVICE%20wikibase%3Alabel%20%7B%20bd%3AserviceParam%20wikibase%3Alanguage%20%22en%22.%20%7D%0A%7D%0AORDER%20BY%20%3FcountryLabel
 |website=Wikidata Query Service
 |access-date=2026-04-12
}}&lt;/ref&gt;

{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Area (km²)
|-
| 1 || Brazil || 8,515,767
|-
| 2 || Argentina || 2,780,400
|-
| 3 || Peru || 1,285,216
|-
| 4 || Colombia || 1,141,748
|-
| 5 || Bolivia || 1,098,581
|}

== Analysis: What the two top-5 lists argue ==

The two rankings suggest that territorial size and population are related in South America, but they do not perfectly match. Brazil ranks first on both lists, which argues that it dominates the continent in both land area and population.&lt;ref name=&quot;combined&quot; /&gt;

The overlap between the two lists is substantial but incomplete. Brazil, Argentina, Peru, and Colombia appear on both lists, suggesting that these countries have both territorial and demographic weight within the continent.&lt;ref name=&quot;combined&quot; /&gt;

The differences are just as important. Venezuela appears in the top 5 by population but not in the top 5 by area, which suggests that a country can be demographically significant without being among the largest by land area. Bolivia appears in the top 5 by area but not in the top 5 by population, which suggests that large territory does not necessarily mean a proportionally large population.&lt;ref name=&quot;combined&quot; /&gt;

Together, the two lists argue that South America has one clear continental giant—Brazil—followed by a second tier of countries whose importance depends on whether the reader is measuring people or territory. The rankings therefore show that geographic scale and human concentration are connected, but they are not identical forms of significance.&lt;ref name=&quot;combined&quot; /&gt;

== Data quality ==

Several data-quality issues should be kept in mind when reading these tables.

* The values come from Wikidata, an open and collaboratively edited database, so figures can change over time as entries are updated.&lt;ref name=&quot;combined&quot; /&gt;
* Population figures may reflect estimates from different years, even when they are queried together in one result set.
* Area values are generally more stable than population values, but formatting can vary. For example, Ecuador appears with a decimal value of 257,204.27 km² while most other countries appear as whole numbers.
* The query is limited to items classified as sovereign states on the continent of South America, so it excludes non-sovereign territories and dependencies.
* The dataset is best understood as a comparative snapshot rather than a fixed official statistical publication.&lt;ref name=&quot;combined&quot; /&gt;

== References ==
&lt;references /&gt;

{{:DesignWriteStudio/SiteElements/Footer}}

[[Category:South America]]</text>
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    <title>DesignWriteStudio/Course/StudentPages/ChrisP/5.2 South America</title>
    <ns>0</ns>
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      <id>2804430</id>
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      <comment>Created page with &quot;{{:DesignWriteStudio/Course/StudentPages/ChrisP/5.1_South_America}}  # Tables Designed to Be Read Interactively  A synthesis essay for Assignment 4.3: Hypertextualizing Data  This page extends the South America dataset by rebuilding the tables as interactive reading interfaces. The country values stay the same, but the structure changes how the reader encounters them. Sortability, wikilinks, collapsibility, and transclusion turn the dataset from a static display into a h...&quot;</comment>
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# Tables Designed to Be Read Interactively

A synthesis essay for Assignment 4.3: Hypertextualizing Data

This page extends the South America dataset by rebuilding the tables as interactive reading interfaces. The country values stay the same, but the structure changes how the reader encounters them. Sortability, wikilinks, collapsibility, and transclusion turn the dataset from a static display into a hypertextual object.&lt;ref&gt;Vandendorpe, C. (2009). ''From Papyrus to Hypertext: Toward the Universal Digital Library''. University of Illinois Press. https://doi.org/10.5406/j.ctt1xcmrr&lt;/ref&gt;&lt;ref&gt;Nelson, T. H. (1965). ''Complex information processing: a file structure for the complex, the changing and the indeterminate''. ACM '65 Proceedings. https://doi.org/10.1145/800197.806036&lt;/ref&gt;&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

== Rebuilt interactive tables ==

=== Complete table of sovereign states ===

The full comparative dataset is placed in a collapsed section so the page preserves access to all countries without overwhelming the opening view.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

&lt;div class=&quot;mw-collapsible mw-collapsed&quot;&gt;
{| class=&quot;wikitable sortable&quot;
! Country
! Population
! Area (km²)
|-
| [[w:Argentina]]
| 47,327,407
| 2,780,400
|-
| [[w:Bolivia]]
| 12,244,159
| 1,098,581
|-
| [[w:Brazil]]
| 213,421,037
| 8,515,767
|-
| [[w:Chile]]
| 19,458,000
| 756,102
|-
| [[w:Colombia]]
| 52,321,152
| 1,141,748
|-
| [[w:Ecuador]]
| 16,938,986
| 257,204.27
|-
| [[w:Guyana]]
| 878,674
| 214,970
|-
| [[w:Paraguay]]
| 6,811,297
| 406,756
|-
| [[w:Peru]]
| 33,726,000
| 1,285,216
|-
| [[w:Suriname]]
| 563,402
| 163,270
|-
| [[w:Uruguay]]
| 3,444,263
| 176,215
|-
| [[w:Venezuela]]
| 31,250,306
| 912,050
|}
&lt;/div&gt;

=== Top 5 by population ===

The largest sovereign states in South America by population are the following.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

&lt;div class=&quot;mw-collapsible&quot;&gt;
{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Population
|-
| 1
| [[w:Brazil]]
| 213,421,037
|-
| 2
| [[w:Colombia]]
| 52,321,152
|-
| 3
| [[w:Argentina]]
| 47,327,407
|-
| 4
| [[w:Peru]]
| 33,726,000
|-
| 5
| [[w:Venezuela]]
| 31,250,306
|}
&lt;/div&gt;

=== Top 5 by area ===

The largest sovereign states in South America by area are the following.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

&lt;div class=&quot;mw-collapsible&quot;&gt;
{| class=&quot;wikitable sortable&quot;
! Rank
! Country
! Area (km²)
|-
| 1
| [[w:Brazil]]
| 8,515,767
|-
| 2
| [[w:Argentina]]
| 2,780,400
|-
| 3
| [[w:Peru]]
| 1,285,216
|-
| 4
| [[w:Colombia]]
| 1,141,748
|-
| 5
| [[w:Bolivia]]
| 1,098,581
|}
&lt;/div&gt;

== Analysis: What the two top-5 lists argue ==

The two rankings suggest that territorial size and population are related in South America, but they do not perfectly match. Brazil ranks first on both lists, which argues that it dominates the continent in both land area and population.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

The overlap between the two lists is substantial but incomplete. Brazil, Argentina, Peru, and Colombia appear on both lists, suggesting that these countries have both territorial and demographic weight within the continent.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

The differences are just as important. Venezuela appears in the top 5 by population but not in the top 5 by area, which suggests that a country can be demographically significant without being among the largest by land area. Bolivia appears in the top 5 by area but not in the top 5 by population, which suggests that large territory does not necessarily mean a proportionally large population.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

Together, the two lists argue that South America has one clear continental giant—Brazil—followed by a second tier of countries whose importance depends on whether the reader is measuring people or territory. The rankings therefore show that geographic scale and human concentration are connected, but they are not identical forms of significance.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

== Data quality ==

Several data-quality issues should be kept in mind when reading these tables.&lt;ref&gt;''Wikidata Query Service: South American sovereign states with population and area''. Wikidata Query Service. Retrieved 2026-04-12. https://query.wikidata.org/&lt;/ref&gt;

* The values come from Wikidata, an open and collaboratively edited database, so figures can change over time as entries are updated.
* Population figures may reflect estimates from different years, even when they are queried together in one result set.
* Area values are generally more stable than population values, but formatting can vary. For example, Ecuador appears with a decimal value of 257,204.27 km² while most other countries appear as whole numbers.
* The query is limited to items classified as sovereign states on the continent of South America, so it excludes non-sovereign territories and dependencies.
* The dataset is best understood as a comparative snapshot rather than a fixed official statistical publication.

== Synthesis ==

Christian Vandendorpe distinguishes between linear and tabular reading. Linear reading follows a sequence determined by the author, while tabular reading is guided by the reader’s purpose.&lt;ref&gt;Vandendorpe, C. (2009). ''From Papyrus to Hypertext: Toward the Universal Digital Library''. University of Illinois Press. https://doi.org/10.5406/j.ctt1xcmrr&lt;/ref&gt; These rebuilt tables support tabular reading because the reader can sort columns, compare rows, and move through the data according to their own question rather than a fixed sequence.

Sortable tables make the page interactive by turning each column header into a new point of entry. The same data can support multiple arguments depending on whether the reader sorts by country, population, or area.&lt;ref&gt;Nelson, T. H. (1965). ''Complex information processing: a file structure for the complex, the changing and the indeterminate''. ACM '65 Proceedings. https://doi.org/10.1145/800197.806036&lt;/ref&gt; Wikilinks then open the table outward, since every country name becomes a gateway to a larger record rather than a closed label. Collapsible sections organize depth, keeping the top-5 rankings visible while hiding the full table until the reader chooses to expand it. Finally, transclusion preserves the 5.1 page as the source of truth while allowing this page to build on it in a new context.&lt;ref&gt;Nelson, T. H. (1995). ''The heart of connection: Hypermedia unified by transclusion''. ''Communications of the ACM'', 38(9), 31–33. https://doi.org/10.1145/223248.223257&lt;/ref&gt;

Taken together, these features show that hypertextualizing data is not just about presentation. It is about restructuring information so the reader can navigate, compare, and interpret it more actively.

== Short reflection ==

Rebuilding the South America tables this way made the dataset feel less like a report and more like a system. The sortable columns invite comparison, the country links connect each row to a wider network of knowledge, and the collapsible sections let the reader decide how much detail they want at a given moment. Transclusion also matters because it keeps the earlier page authoritative instead of duplicating it. The result is a page that does not simply display data, but encourages the reader to interact with it.

== References ==
&lt;references /&gt;</text>
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    <title>User talk:14gulaman</title>
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        <id>10242</id>
      </contributor>
      <comment>Welcome</comment>
      <origin>2804457</origin>
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      <format>text/x-wiki</format>
      <text bytes="2326" sha1="9gmtmonazka63d5jae6p8hy936isdmn" xml:space="preserve">==Welcome==
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To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:16, 12 April 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
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      <comment>I have added a welcome message and the purpose of the page.</comment>
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      <text bytes="312" sha1="j0ac7ohusguhione24wfzw0zakibob6" xml:space="preserve">Welcome to a dialogical space dedicated to discussions about a 2026 essay titled &quot;xxxx&quot;, published in xxxx by xxxx (to be updated once the essay is published). 

Here you can add reflections on the essay, ask the author questions, or request/provide resources. A direct link to the essay is available here: xxxx.</text>
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      <comment>Welcome</comment>
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      <format>text/x-wiki</format>
      <text bytes="2323" sha1="0q4bfqcskt4rlnjvj1zb7fzu7aho80i" xml:space="preserve">==Welcome==
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You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Jtneill|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

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&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
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* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
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To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! ---- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:15, 12 April 2026 (UTC)&lt;/div&gt;
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